Quadratic equations often appear when solving various problems in physics and mathematics. In this article we will look at how to solve these equalities in a universal way “through a discriminant”. Examples of using the acquired knowledge are also given in the article.

What equations will we be talking about?

The figure below shows a formula in which x is an unknown variable and the Latin symbols a, b, c represent some known numbers.

Each of these symbols is called a coefficient. As you can see, the number "a" appears before the variable x squared. This is the maximum power of the expression represented, which is why it is called a quadratic equation. Its other name is often used: second-order equation. The value a itself is a square coefficient (standing when the variable is squared), b is linear coefficient(it is located next to the variable raised to the first power), finally, the number c is the free term.

Note that the type of equation shown in the figure above is a general classical quadratic expression. In addition to it, there are other second-order equations in which the coefficients b and c can be zero.

When the task is set to solve the equality in question, this means that such values ​​of the variable x need to be found that would satisfy it. Here, the first thing you need to remember is the following thing: since the maximum degree of X is 2, then this type of expression cannot have more than 2 solutions. This means that if, when solving an equation, 2 values ​​of x were found that satisfy it, then you can be sure that there is no 3rd number, substituting it for x, the equality would also be true. The solutions to an equation in mathematics are called its roots.

Methods for solving second order equations

Solving equations of this type requires knowledge of some theory about them. In the school algebra course, 4 different solution methods are considered. Let's list them:

• using factorization;
• using the formula for a perfect square;
• by applying the graph of the corresponding quadratic function;
• using the discriminant equation.

The advantage of the first method is its simplicity; however, it cannot be used for all equations. The second method is universal, but somewhat cumbersome. The third method is distinguished by its clarity, but it is not always convenient and applicable. And finally, using the discriminant equation is a universal and fairly simple way to find the roots of absolutely any second-order equation. Therefore, in this article we will consider only it.

Formula for obtaining the roots of the equation

Let us turn to the general form of the quadratic equation. Let's write it down: a*x²+ b*x + c =0. Before using the method of solving it “through a discriminant,” you should always bring the equality to its written form. That is, it must consist of three terms (or less if b or c is 0).

For example, if there is an expression: x²-9*x+8 = -5*x+7*x², then you should first move all its terms to one side of the equality and add the terms containing the variable x in the same powers.

In this case, this operation will lead to the following expression: -6*x²-4*x+8=0, which is equivalent to the equation 6*x²+4*x-8=0 (here we multiplied the left and right sides of the equality by -1) .

In the example above, a = 6, b=4, c=-8. Note that all terms of the equality under consideration are always summed together, so if the “-” sign appears, this means that the corresponding coefficient is negative, like the number c in this case.

Having examined this point, let us now move on to the formula itself, which makes it possible to obtain the roots of a quadratic equation. It looks like the one shown in the photo below.

As can be seen from this expression, it allows you to get two roots (pay attention to the “±” sign). To do this, it is enough to substitute the coefficients b, c, and a into it.

The concept of a discriminant

In the previous paragraph, a formula was given that allows you to quickly solve any second-order equation. In it, the radical expression is called a discriminant, that is, D = b²-4*a*c.

Why is this part of the formula highlighted, and it even has proper name? The fact is that the discriminant connects all three coefficients of the equation into a single expression. Last fact means that it completely carries information about the roots, which can be expressed in the following list:

1. D>0: equality has 2 various solutions, both of which are real numbers.
2. D=0: The equation has only one root, and it is a real number.

Discriminant determination task

Let's give a simple example of how to find a discriminant. Let the following equality be given: 2*x² - 4+5*x-9*x² = 3*x-5*x²+7.

Let's bring it to standard form, we get: (2*x²-9*x²+5*x²) + (5*x-3*x) + (- 4-7) = 0, from which we come to the equality: -2*x² +2*x-11 = 0. Here a=-2, b=2, c=-11.

Now you can use the above formula for the discriminant: D = 2² - 4*(-2)*(-11) = -84. The resulting number is the answer to the task. Since in the example the discriminant less than zero, then we can say that this quadratic equation has no real roots. Its solution will be only numbers of complex type.

An example of inequality through a discriminant

Let's solve problems of a slightly different type: given the equality -3*x²-6*x+c = 0. It is necessary to find values ​​of c for which D>0.

In this case, only 2 out of 3 coefficients are known, so it is not possible to calculate the exact value of the discriminant, but it is known that it is positive. We use the last fact when composing the inequality: D= (-6)²-4*(-3)*c>0 => 36+12*c>0. Solving the resulting inequality leads to the result: c>-3.

Let's check the resulting number. To do this, we calculate D for 2 cases: c=-2 and c=-4. The number -2 satisfies the obtained result (-2>-3), the corresponding discriminant will have the value: D = 12>0. In turn, the number -4 does not satisfy the inequality (-4. Thus, any numbers c that are greater than -3 will satisfy the condition.

An example of solving an equation

Let us present a problem that involves not only finding the discriminant, but also solving the equation. It is necessary to find the roots for the equality -2*x²+7-9*x = 0.

In this example, the discriminant is equal to the following value: D = 81-4*(-2)*7= 137. Then the roots of the equation are determined as follows: x = (9±√137)/(-4). This exact values roots, if you calculate the root approximately, then you get the numbers: x = -5.176 and x = 0.676.

Geometric problem

We will solve a problem that will require not only the ability to calculate the discriminant, but also the application of skills abstract thinking and knowledge of how to write quadratic equations.

Bob had a 5 x 4 meter duvet. The boy wanted to sew it around the entire perimeter continuous strip from beautiful fabric. How thick will this strip be if we know that Bob has 10 m² of fabric.

Let the strip have a thickness of x m, then the area of ​​the fabric along the long side of the blanket will be (5+2*x)*x, and since there are 2 long sides, we have: 2*x*(5+2*x). On the short side, the area of ​​​​the sewn fabric will be 4*x, since there are 2 of these sides, we get the value 8*x. Note that the value 2*x was added to the long side because the length of the blanket increased by that number. The total area of ​​fabric sewn to the blanket is 10 m². Therefore, we get the equality: 2*x*(5+2*x) + 8*x = 10 => 4*x²+18*x-10 = 0.

For this example, the discriminant is equal to: D = 18²-4*4*(-10) = 484. Its root is 22. Using the formula, we find the required roots: x = (-18±22)/(2*4) = (- 5; 0.5). Obviously, of the two roots, only the number 0.5 is suitable according to the conditions of the problem.

Thus, the strip of fabric that Bob sews to his blanket will be 50 cm wide.

Quadratic equation - easy to solve! *Hereinafter referred to as “KU”. Friends, it would seem that there could be nothing simpler in mathematics than solving such an equation. But something told me that many people have problems with him. I decided to see how many on-demand impressions Yandex gives out per month. Here's what happened, look:

What does it mean? This means that about 70,000 people a month are looking for this information, what does this summer have to do with it, and what will happen among school year— there will be twice as many requests. This is not surprising, because those guys and girls who graduated from school a long time ago and are preparing for the Unified State Exam are looking for this information, and schoolchildren also strive to refresh their memory.

Despite the fact that there are a lot of sites that tell you how to solve this equation, I decided to also contribute and publish the material. Firstly, I want visitors to come to my site based on this request; secondly, in other articles, when the topic of “KU” comes up, I will provide a link to this article; thirdly, I’ll tell you a little more about his solution than is usually stated on other sites. Let's get started! The content of the article:

A quadratic equation is an equation of the form:

where coefficients a,band c are arbitrary numbers, with a≠0.

In the school course, the material is given in the following form - the equations are divided into three classes:

1. They have two roots.

2. *Have only one root.

3. They have no roots. It is worth especially noting here that they do not have real roots

How are roots calculated? Just!

We calculate the discriminant. Underneath this “terrible” word lies a very simple formula:

The root formulas are as follows:

*You need to know these formulas by heart.

You can immediately write down and solve:

Example:

1. If D > 0, then the equation has two roots.

2. If D = 0, then the equation has one root.

3. If D< 0, то уравнение не имеет действительных корней.

Let's look at the equation:

In this regard, when the discriminant is equal to zero, the school course says that one root is obtained, here it is equal to nine. Everything is correct, it is so, but...

This idea is somewhat incorrect. In fact, there are two roots. Yes, yes, don’t be surprised, you get two equal roots, and to be mathematically precise, then the answer should write two roots:

x 1 = 3 x 2 = 3

But this is so - a small digression. At school you can write it down and say that there is one root.

Now the next example:

As we know, the root of a negative number cannot be taken, so there is no solution in this case.

That's the whole decision process.

Quadratic function.

This shows what the solution looks like geometrically. This is extremely important to understand (in the future, in one of the articles we will analyze in detail the solution to the quadratic inequality).

This is a function of the form:

where x and y are variables

a, b, c – given numbers, with a ≠ 0

The graph is a parabola:

That is, it turns out that by solving a quadratic equation with “y” equal to zero, we find the points of intersection of the parabola with the x axis. There can be two of these points (the discriminant is positive), one (the discriminant is zero) and none (the discriminant is negative). Details about the quadratic function You can view article by Inna Feldman.

Let's look at examples:

Example 1: Solve 2x 2 +8 x–192=0

a=2 b=8 c= –192

D=b 2 –4ac = 8 2 –4∙2∙(–192) = 64+1536 = 1600

Answer: x 1 = 8 x 2 = –12

*It was possible to immediately divide the left and right sides of the equation by 2, that is, simplify it. The calculations will be easier.

Example 2: Decide x 2–22 x+121 = 0

a=1 b=–22 c=121

D = b 2 –4ac =(–22) 2 –4∙1∙121 = 484–484 = 0

We found that x 1 = 11 and x 2 = 11

It is permissible to write x = 11 in the answer.

Answer: x = 11

Example 3: Decide x 2 –8x+72 = 0

a=1 b= –8 c=72

D = b 2 –4ac =(–8) 2 –4∙1∙72 = 64–288 = –224

The discriminant is negative, there is no solution in real numbers.

Answer: no solution

The discriminant is negative. There is a solution!

Here we will talk about solving the equation in the case when a negative discriminant is obtained. Do you know anything about complex numbers? I will not go into detail here about why and where they arose and what their specific role and necessity in mathematics is; this is a topic for a large separate article.

The concept of a complex number.

A little theory.

A complex number z is a number of the form

z = a + bi

where a and b are real numbers, i is the so-called imaginary unit.

a+bi – this is a SINGLE NUMBER, not an addition.

The imaginary unit is equal to the root of minus one:

Now consider the equation:

We get two conjugate roots.

Incomplete quadratic equation.

Let's consider special cases, this is when the coefficient “b” or “c” is equal to zero (or both are equal to zero). They can be solved easily without any discriminants.

Case 1. Coefficient b = 0.

The equation becomes:

Let's transform:

Example:

4x 2 –16 = 0 => 4x 2 =16 => x 2 = 4 => x 1 = 2 x 2 = –2

Case 2. Coefficient c = 0.

The equation becomes:

Let's transform and factorize:

*The product is equal to zero when at least one of the factors is equal to zero.

Example:

9x 2 –45x = 0 => 9x (x–5) =0 => x = 0 or x–5 =0

x 1 = 0 x 2 = 5

Case 3. Coefficients b = 0 and c = 0.

Here it is clear that the solution to the equation will always be x = 0.

Useful properties and patterns of coefficients.

There are properties that allow you to solve equations with large coefficients.

Ax 2 + bx+ c=0 equality holds

a + b+ c = 0, That

- if for the coefficients of the equation Ax 2 + bx+ c=0 equality holds

a+ c =b, That

These properties help solve a certain type of equation.

Example 1: 5001 x 2 –4995 x – 6=0

The sum of the odds is 5001+( – 4995)+(– 6) = 0, which means

Example 2: 2501 x 2 +2507 x+6=0

Equality holds a+ c =b, Means

Regularities of coefficients.

1. If in the equation ax 2 + bx + c = 0 the coefficient “b” is equal to (a 2 +1), and the coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 + (a 2 +1)∙x+ a= 0 = > x 1 = –a x 2 = –1/a.

Example. Consider the equation 6x 2 + 37x + 6 = 0.

x 1 = –6 x 2 = –1/6.

2. If in the equation ax 2 – bx + c = 0 the coefficient “b” is equal to (a 2 +1), and the coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 – (a 2 +1)∙x+ a= 0 = > x 1 = a x 2 = 1/a.

Example. Consider the equation 15x 2 –226x +15 = 0.

x 1 = 15 x 2 = 1/15.

3. If in Eq. ax 2 + bx – c = 0 coefficient “b” is equal to (a 2 – 1), and coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 + (a 2 –1)∙x – a= 0 = > x 1 = – a x 2 = 1/a.

Example. Consider the equation 17x 2 +288x – 17 = 0.

x 1 = – 17 x 2 = 1/17.

4. If in the equation ax 2 – bx – c = 0 the coefficient “b” is equal to (a 2 – 1), and the coefficient c is numerically equal to the coefficient “a”, then its roots are equal

ax 2 – (a 2 –1)∙x – a= 0 = > x 1 = a x 2 = – 1/a.

Example. Consider the equation 10x 2 – 99x –10 = 0.

x 1 = 10 x 2 = – 1/10

Vieta's theorem.

Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, we can express the sum and product of the roots of an arbitrary KU in terms of its coefficients.

45 = 1∙45 45 = 3∙15 45 = 5∙9.

In total, the number 14 gives only 5 and 9. These are roots. With a certain skill, using the presented theorem, you can solve many quadratic equations orally immediately.

Vieta's theorem, in addition. It is convenient in that after solving a quadratic equation in the usual way (through a discriminant), the resulting roots can be checked. I recommend doing this always.

TRANSPORTATION METHOD

With this method, the coefficient “a” is multiplied by the free term, as if “thrown” to it, which is why it is called "transfer" method. This method is used when the roots of the equation can be easily found using Vieta's theorem and, most importantly, when the discriminant is an exact square.

If A± b+c≠ 0, then the transfer technique is used, for example:

2X 2 – 11x+ 5 = 0 (1) => X 2 – 11x+ 10 = 0 (2)

Using Vieta's theorem in equation (2), it is easy to determine that x 1 = 10 x 2 = 1

The resulting roots of the equation must be divided by 2 (since the two were “thrown” from x 2), we get

x 1 = 5 x 2 = 0.5.

What is the rationale? Look what's happening.

The discriminants of equations (1) and (2) are equal:

If you look at the roots of the equations, you only get different denominators, and the result depends precisely on the coefficient of x 2:

The second (modified) one has roots that are 2 times larger.

Therefore, we divide the result by 2.

*If we reroll the three, we will divide the result by 3, etc.

Answer: x 1 = 5 x 2 = 0.5

Sq. ur-ie and Unified State Examination.

I’ll tell you briefly about its importance - YOU MUST BE ABLE TO DECIDE quickly and without thinking, you need to know the formulas of roots and discriminants by heart. Many of the problems included in the Unified State Examination tasks boil down to solving a quadratic equation (geometric ones included).

Something worth noting!

1. The form of writing an equation can be “implicit”. For example, the following entry is possible:

15+ 9x 2 - 45x = 0 or 15x+42+9x 2 - 45x=0 or 15 -5x+10x 2 = 0.

You need to bring it to a standard form (so as not to get confused when solving).

2. Remember that x is an unknown quantity and it can be denoted by any other letter - t, q, p, h and others.

First level

Quadratic equations. Comprehensive guide (2019)

In the term “quadratic equation,” the key word is “quadratic.” This means that the equation must necessarily contain a variable (that same x) squared, and there should not be xes to the third (or greater) power.

The solution of many equations comes down to solving quadratic equations.

Let's learn to determine that this is a quadratic equation and not some other equation.

Example 1.

Let's get rid of the denominator and multiply each term of the equation by

Let's move everything to left side and arrange the terms in descending order of powers of x

Now we can say with confidence that given equation is square!

Example 2.

Multiply the left and right sides by:

This equation, although it was originally in it, is not quadratic!

Example 3.

Let's multiply everything by:

Scary? The fourth and second degrees... However, if we make a replacement, we will see that we have a simple quadratic equation:

Example 4.

It seems to be there, but let's take a closer look. Let's move everything to the left side:

See, it's reduced - and now it's a simple linear equation!

Now try to determine for yourself which of the following equations are quadratic and which are not:

Examples:

Answers:

1. square;
2. square;
3. not square;
4. not square;
5. not square;
6. square;
7. not square;
8. square.

Mathematicians conventionally divide all quadratic equations into the following types:

• Complete quadratic equations- equations in which the coefficients and, as well as the free term c, are not equal to zero (as in the example). In addition, among complete quadratic equations there are given- these are equations in which the coefficient (the equation from example one is not only complete, but also reduced!)

• Incomplete quadratic equations- equations in which the coefficient and or the free term c are equal to zero:

  They are incomplete because they are missing some element. But the equation must always contain x squared!!! Otherwise, it will no longer be a quadratic equation, but some other equation.

Why did they come up with such a division? It would seem that there is an X squared, and okay. This division is determined by the solution methods. Let's look at each of them in more detail.

Solving incomplete quadratic equations

First, let's focus on solving incomplete quadratic equations - they are much simpler!

There are types of incomplete quadratic equations:

1. , in this equation the coefficient is equal.
2. , in this equation the free term is equal to.
3. , in this equation the coefficient and the free term are equal.

1. i. Because we know how to extract Square root, then let's express from this equation

The expression can be either negative or positive. A squared number cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number, so: if, then the equation has no solutions.

And if, then we get two roots. There is no need to memorize these formulas. The main thing is that you must know and always remember that it cannot be less.

Let's try to solve some examples.

Example 5:

Solve the equation

Now all that remains is to extract the root from the left and right sides. After all, you remember how to extract roots?

Answer:

Never forget about roots with a negative sign!!!

Example 6:

Solve the equation

Answer:

Example 7:

Solve the equation

Oh! The square of a number cannot be negative, which means that the equation

no roots!

For such equations that have no roots, mathematicians came up with a special icon - (empty set). And the answer can be written like this:

Answer:

Thus, this quadratic equation has two roots. There are no restrictions here, since we did not extract the root.
Example 8:

Solve the equation

Let's take the common factor out of brackets:

Thus,

This equation has two roots.

Answer:

The simplest type of incomplete quadratic equations (although they are all simple, right?). Obviously, this equation always has only one root:

We will dispense with examples here.

Solving complete quadratic equations

We remind you that a complete quadratic equation is an equation of the form equation where

Solving complete quadratic equations is a little more difficult (just a little) than these.

Remember, Any quadratic equation can be solved using a discriminant! Even incomplete.

The other methods will help you do it faster, but if you have problems with quadratic equations, first master the solution using the discriminant.

1. Solving quadratic equations using a discriminant.

Solving quadratic equations using this method is very simple; the main thing is to remember the sequence of actions and a couple of formulas.

If, then the equation has a root. Special attention take a step. Discriminant () tells us the number of roots of the equation.

• If, then the formula in the step will be reduced to. Thus, the equation will only have a root.
• If, then we will not be able to extract the root of the discriminant at the step. This indicates that the equation has no roots.

Let's go back to our equations and look at some examples.

Example 9:

Solve the equation

Step 1 we skip.

Step 2.

We find the discriminant:

This means the equation has two roots.

Step 3.

Answer:

Example 10:

Solve the equation

The equation is presented in standard form, so Step 1 we skip.

Step 2.

We find the discriminant:

This means that the equation has one root.

Answer:

Example 11:

Solve the equation

The equation is presented in standard form, so Step 1 we skip.

Step 2.

We find the discriminant:

This means we will not be able to extract the root of the discriminant. There are no roots of the equation.

Now we know how to correctly write down such answers.

Answer: no roots

2. Solving quadratic equations using Vieta's theorem.

If you remember, there is a type of equation that is called reduced (when the coefficient a is equal to):

Such equations are very easy to solve using Vieta’s theorem:

Sum of roots given quadratic equation is equal, and the product of the roots is equal.

Example 12:

Solve the equation

This equation can be solved using Vieta's theorem because .

The sum of the roots of the equation is equal, i.e. we get the first equation:

And the product is equal to:

Let's compose and solve the system:

• And. The amount is equal to;
• And. The amount is equal to;
• And. The amount is equal.

and are the solution to the system:

Answer: ; .

Example 13:

Solve the equation

Answer:

Example 14:

Solve the equation

The equation is given, which means:

Answer:

QUADRATIC EQUATIONS. AVERAGE LEVEL

What is a quadratic equation?

In other words, a quadratic equation is an equation of the form, where - the unknown, - some numbers, and.

The number is called the highest or first coefficient quadratic equation, - second coefficient, A - free member.

Why? Because if the equation immediately becomes linear, because will disappear.

In this case, and can be equal to zero. In this chair equation is called incomplete. If all the terms are in place, that is, the equation is complete.

Solutions to various types of quadratic equations

Methods for solving incomplete quadratic equations:

First, let's look at methods for solving incomplete quadratic equations - they are simpler.

We can distinguish the following types of equations:

I., in this equation the coefficient and the free term are equal.

II. , in this equation the coefficient is equal.

III. , in this equation the free term is equal to.

Now let's look at the solution to each of these subtypes.

Obviously, this equation always has only one root:

A squared number cannot be negative, because when you multiply two negative or two positive numbers, the result will always be a positive number. That's why:

if, then the equation has no solutions;

if we have two roots

There is no need to memorize these formulas. The main thing to remember is that it cannot be less.

Examples:

Solutions:

Answer:

Never forget about roots with a negative sign!

The square of a number cannot be negative, which means that the equation

no roots.

To briefly write down that a problem has no solutions, we use the empty set icon.

Answer:

So, this equation has two roots: and.

Answer:

Let's take the common factor out of brackets:

The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:

So, this quadratic equation has two roots: and.

Example:

Solve the equation.

Solution:

Let's factor the left side of the equation and find the roots:

Answer:

Methods for solving complete quadratic equations:

1. Discriminant

Solving quadratic equations this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using a discriminant! Even incomplete.

Did you notice the root from the discriminant in the formula for roots? But the discriminant can be negative. What to do? We need to pay special attention to step 2. The discriminant tells us the number of roots of the equation.

• If, then the equation has roots:
• If, then the equation has the same roots, and in fact, one root:

  Such roots are called double roots.

• If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.

Why is it possible different quantities roots? Let's turn to geometric sense quadratic equation. The graph of the function is a parabola:

In a special case, which is a quadratic equation, . This means that the roots of a quadratic equation are the points of intersection with the abscissa axis (axis). A parabola may not intersect the axis at all, or may intersect it at one (when the vertex of the parabola lies on the axis) or two points.

In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upward, and if, then downward.

Examples:

Solutions:

Answer:

Answer: .

Answer:

This means there are no solutions.

Answer: .

2. Vieta's theorem

It is very easy to use Vieta's theorem: you just need to choose a pair of numbers whose product is equal to the free term of the equation, and the sum is equal to the second coefficient taken with the opposite sign.

It is important to remember that Vieta's theorem can only be applied in reduced quadratic equations ().

Let's look at a few examples:

Example #1:

Solve the equation.

Solution:

This equation can be solved using Vieta's theorem because . Other coefficients: ; .

The sum of the roots of the equation is:

And the product is equal to:

Let's select pairs of numbers whose product is equal and check whether their sum is equal:

• And. The amount is equal to;
• And. The amount is equal to;
• And. The amount is equal.

and are the solution to the system:

Thus, and are the roots of our equation.

Answer: ; .

Example #2:

Solution:

Let's select pairs of numbers that give in the product, and then check whether their sum is equal:

and: they give in total.

and: they give in total. To obtain, it is enough to simply change the signs of the supposed roots: and, after all, the product.

Answer:

Example #3:

Solution:

The free term of the equation is negative, and therefore the product of the roots is a negative number. This is only possible if one of the roots is negative and the other is positive. Therefore the sum of the roots is equal to differences of their modules.

Let us select pairs of numbers that give in the product, and whose difference is equal to:

and: their difference is equal - does not fit;

and: - not suitable;

and: - not suitable;

and: - suitable. All that remains is to remember that one of the roots is negative. Since their sum must be equal, the root with the smaller modulus must be negative: . We check:

Answer:

Example #4:

Solve the equation.

Solution:

The equation is given, which means:

The free term is negative, and therefore the product of the roots is negative. And this is only possible when one root of the equation is negative and the other is positive.

Let's select pairs of numbers whose product is equal, and then determine which roots should have a negative sign:

Obviously, only the roots and are suitable for the first condition:

Answer:

Example #5:

Solve the equation.

Solution:

The equation is given, which means:

The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, it means both roots have a minus sign.

Let us select pairs of numbers whose product is equal to:

Obviously, the roots are the numbers and.

Answer:

Agree, it’s very convenient to come up with roots orally, instead of counting this nasty discriminant. Try to use Vieta's theorem as often as possible.

But Vieta’s theorem is needed in order to facilitate and speed up finding the roots. In order for you to benefit from using it, you must bring the actions to automaticity. And for this, solve five more examples. But don't cheat: you can't use a discriminant! Only Vieta's theorem:

Solutions to tasks for independent work:

Task 1. ((x)^(2))-8x+12=0

According to Vieta's theorem:

As usual, we start the selection with the piece:

Not suitable because the amount;

: the amount is just what you need.

Answer: ; .

Task 2.

And again our favorite Vieta theorem: the sum must be equal, and the product must be equal.

But since it must be not, but, we change the signs of the roots: and (in total).

Answer: ; .

Task 3.

Hmm... Where is that?

You need to move all the terms into one part:

The sum of the roots is equal to the product.

Okay, stop! The equation is not given. But Vieta's theorem is applicable only in the given equations. So first you need to give an equation. If you can’t lead, give up this idea and solve it in another way (for example, through a discriminant). Let me remind you that to give a quadratic equation means to make the leading coefficient equal:

Great. Then the sum of the roots is equal to and the product.

Here it’s as easy as shelling pears to choose: after all, it’s a prime number (sorry for the tautology).

Answer: ; .

Task 4.

The free member is negative. What's special about this? And the fact is that the roots will have different signs. And now, during the selection, we check not the sum of the roots, but the difference in their modules: this difference is equal, but a product.

So, the roots are equal to and, but one of them is minus. Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is. This means that the smaller root will have a minus: and, since.

Answer: ; .

Task 5.

What should you do first? That's right, give the equation:

Again: we select the factors of the number, and their difference should be equal to:

The roots are equal to and, but one of them is minus. Which? Their sum should be equal, which means that the minus will have a larger root.

Answer: ; .

Let me summarize:

1. Vieta's theorem is used only in the quadratic equations given.
2. Using Vieta's theorem, you can find the roots by selection, orally.
3. If the equation is not given or no equation is found suitable pair multipliers of the free term, which means there are no whole roots, and you need to solve it in another way (for example, through a discriminant).

3. Method for selecting a complete square

If all terms containing the unknown are represented in the form of terms from abbreviated multiplication formulas - the square of the sum or difference - then after replacing variables, the equation can be presented in the form of an incomplete quadratic equation of the type.

For example:

Example 1:

Solve the equation: .

Solution:

Answer:

Example 2:

Solve the equation: .

Solution:

Answer:

IN general view the transformation will look like this:

This implies: .

Doesn't remind you of anything? This is a discriminatory thing! That's exactly how we got the discriminant formula.

QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN THINGS

Quadratic equation- this is an equation of the form, where - the unknown, - the coefficients of the quadratic equation, - the free term.

Complete quadratic equation- an equation in which the coefficients are not equal to zero.

Reduced quadratic equation- an equation in which the coefficient, that is: .

Incomplete quadratic equation- an equation in which the coefficient and or the free term c are equal to zero:

• if the coefficient, the equation looks like: ,
• if there is a free term, the equation has the form: ,
• if and, the equation looks like: .

1. Algorithm for solving incomplete quadratic equations

1.1. An incomplete quadratic equation of the form, where, :

1) Let's express the unknown: ,

2) Check the sign of the expression:

• if, then the equation has no solutions,
• if, then the equation has two roots.

1.2. An incomplete quadratic equation of the form, where, :

1) Let’s take the common factor out of brackets: ,

2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:

1.3. An incomplete quadratic equation of the form, where:

This equation always has only one root: .

2. Algorithm for solving complete quadratic equations of the form where

2.1. Solution using discriminant

1) Let's bring the equation to standard form: ,

2) Let's calculate the discriminant using the formula: , which indicates the number of roots of the equation:

3) Find the roots of the equation:

• if, then the equation has roots, which are found by the formula:
• if, then the equation has a root, which is found by the formula:
• if, then the equation has no roots.

2.2. Solution using Vieta's theorem

The sum of the roots of the reduced quadratic equation (equation of the form where) is equal, and the product of the roots is equal, i.e. , A.

2.3. Solution by the method of selecting a complete square

If a quadratic equation of the form has roots, then it can be written in the form: .

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Continuing the topic “Solving Equations,” the material in this article will introduce you to quadratic equations.

Let's look at everything in detail: the essence and recording of the quadratic equation, define the associated terms, analyze the scheme for solving incomplete and complete equations, let's get acquainted with the formula of roots and discriminant, establish connections between roots and coefficients, and of course we will give a visual solution to practical examples.

Yandex.RTB R-A-339285-1

Quadratic equation, its types

Definition 1

Quadratic equation is an equation written as a x 2 + b x + c = 0, Where x– variable, a , b and c– some numbers, while a is not zero.

Often, quadratic equations are also called equations of the second degree, since in essence a quadratic equation is algebraic equation second degree.

Let's give an example to illustrate the given definition: 9 x 2 + 16 x + 2 = 0 ; 7, 5 x 2 + 3, 1 x + 0, 11 = 0, etc. These are quadratic equations.

Definition 2

Numbers a, b and c are the coefficients of the quadratic equation a x 2 + b x + c = 0, while the coefficient a is called the first, or senior, or coefficient at x 2, b - the second coefficient, or coefficient at x, A c called a free member.

For example, in the quadratic equation 6 x 2 − 2 x − 11 = 0 the leading coefficient is 6, the second coefficient is − 2 , and the free term is equal to − 11 . Let us pay attention to the fact that when the coefficients b and/or c are negative, then use short form records like 6 x 2 − 2 x − 11 = 0, but not 6 x 2 + (− 2) x + (− 11) = 0.

Let us also clarify this aspect: if the coefficients a and/or b equal 1 or − 1 , then they may not take an explicit part in writing the quadratic equation, which is explained by the peculiarities of writing the indicated numerical coefficients. For example, in the quadratic equation y 2 − y + 7 = 0 the leading coefficient is 1, and the second coefficient is − 1 .

Reduced and unreduced quadratic equations

Based on the value of the first coefficient, quadratic equations are divided into reduced and unreduced.

Definition 3

Reduced quadratic equation is a quadratic equation where the leading coefficient is 1. For other values ​​of the leading coefficient, the quadratic equation is unreduced.

Let's give examples: quadratic equations x 2 − 4 · x + 3 = 0, x 2 − x − 4 5 = 0 are reduced, in each of which the leading coefficient is 1.

9 x 2 − x − 2 = 0- unreduced quadratic equation, where the first coefficient is different from 1 .

Any unreduced quadratic equation can be converted into a reduced equation by dividing both sides by the first coefficient (equivalent transformation). The transformed equation will have the same roots as the given unreduced equation or will also have no roots at all.

Consideration concrete example will allow us to clearly demonstrate the transition from an unreduced quadratic equation to a reduced one.

Example 1

Given the equation 6 x 2 + 18 x − 7 = 0 . It is necessary to convert the original equation into the reduced form.

Solution

According to the above scheme, we divide both sides of the original equation by the leading coefficient 6. Then we get: (6 x 2 + 18 x − 7) : 3 = 0: 3, and this is the same as: (6 x 2) : 3 + (18 x) : 3 − 7: 3 = 0 and further: (6: 6) x 2 + (18: 6) x − 7: 6 = 0. From here: x 2 + 3 x - 1 1 6 = 0 . Thus, an equation equivalent to the given one is obtained.

Answer: x 2 + 3 x - 1 1 6 = 0 .

Complete and incomplete quadratic equations

Let's turn to the definition of a quadratic equation. In it we specified that a ≠ 0. A similar condition is necessary for the equation a x 2 + b x + c = 0 was precisely square, since at a = 0 it essentially transforms into a linear equation b x + c = 0.

In the case when the coefficients b And c are equal to zero (which is possible, both individually and jointly), the quadratic equation is called incomplete.

Definition 4

Incomplete quadratic equation- such a quadratic equation a x 2 + b x + c = 0, where at least one of the coefficients b And c(or both) is zero.

Complete quadratic equation– a quadratic equation in which all numerical coefficients are not equal to zero.

Let's discuss why the types of quadratic equations are given exactly these names.

When b = 0, the quadratic equation takes the form a x 2 + 0 x + c = 0, which is the same as a x 2 + c = 0. At c = 0 the quadratic equation is written as a x 2 + b x + 0 = 0, which is equivalent a x 2 + b x = 0. At b = 0 And c = 0 the equation will take the form a x 2 = 0. The equations that we obtained differ from the complete quadratic equation in that their left-hand sides do not contain either a term with the variable x, or a free term, or both. Actually, this fact gave the name to this type of equation – incomplete.

For example, x 2 + 3 x + 4 = 0 and − 7 x 2 − 2 x + 1, 3 = 0 are complete quadratic equations; x 2 = 0, − 5 x 2 = 0; 11 x 2 + 2 = 0, − x 2 − 6 x = 0 – incomplete quadratic equations.

Solving incomplete quadratic equations

The definition given above makes it possible to highlight the following types incomplete quadratic equations:

• a x 2 = 0, this equation corresponds to the coefficients b = 0 and c = 0 ;
• a · x 2 + c = 0 at b = 0 ;
• a · x 2 + b · x = 0 at c = 0.

Let us consider sequentially the solution of each type of incomplete quadratic equation.

Solution of the equation a x 2 =0

As mentioned above, this equation corresponds to the coefficients b And c, equal to zero. The equation a x 2 = 0 can be converted into an equivalent equation x 2 = 0, which we get by dividing both sides of the original equation by the number a, not equal to zero. The obvious fact is that the root of the equation x 2 = 0 this is zero because 0 2 = 0 . This equation has no other roots, which can be explained by the properties of the degree: for any number p, not equal to zero, the inequality is true p 2 > 0, from which it follows that when p ≠ 0 equality p 2 = 0 will never be achieved.

Definition 5

Thus, for the incomplete quadratic equation a x 2 = 0 there is a unique root x = 0.

Example 2

For example, let’s solve an incomplete quadratic equation − 3 x 2 = 0. It is equivalent to the equation x 2 = 0, its only root is x = 0, then the original equation has a single root - zero.

Briefly, the solution is written as follows:

− 3 x 2 = 0, x 2 = 0, x = 0.

Solving the equation a x 2 + c = 0

Next in line is the solution of incomplete quadratic equations, where b = 0, c ≠ 0, that is, equations of the form a x 2 + c = 0. Let's transform this equation by moving a term from one side of the equation to the other, changing the sign to the opposite one and dividing both sides of the equation by a number that is not equal to zero:

• transfer c to the right hand side, which gives the equation a x 2 = − c;
• divide both sides of the equation by a, we end up with x = - c a .

Our transformations are equivalent; accordingly, the resulting equation is also equivalent to the original one, and this fact makes it possible to draw conclusions about the roots of the equation. From what the values ​​are a And c the value of the expression - c a depends: it can have a minus sign (for example, if a = 1 And c = 2, then - c a = - 2 1 = - 2) or a plus sign (for example, if a = − 2 And c = 6, then - c a = - 6 - 2 = 3); it is not zero because c ≠ 0. Let us dwell in more detail on situations when - c a< 0 и - c a > 0 .

In the case when - c a< 0 , уравнение x 2 = - c a не будет иметь корней. Утверждая это, мы опираемся на то, что квадратом любого числа является число неотрицательное. Из сказанного следует, что при - c a < 0 ни для какого числа p the equality p 2 = - c a cannot be true.

Everything is different when - c a > 0: remember the square root, and it will become obvious that the root of the equation x 2 = - c a will be the number - c a, since - c a 2 = - c a. It is not difficult to understand that the number - - c a is also the root of the equation x 2 = - c a: indeed, - - c a 2 = - c a.

The equation will have no other roots. We can demonstrate this using the method of contradiction. To begin with, let us define the notations for the roots found above as x 1 And − x 1. Let us assume that the equation x 2 = - c a also has a root x 2, which is different from the roots x 1 And − x 1. We know that by substituting into the equation x its roots, we transform the equation into a fair numerical equality.

For x 1 And − x 1 we write: x 1 2 = - c a , and for x 2- x 2 2 = - c a . Based on the properties of numerical equalities, we subtract one correct equality term by term from another, which will give us: x 1 2 − x 2 2 = 0. We use the properties of operations with numbers to rewrite the last equality as (x 1 − x 2) · (x 1 + x 2) = 0. It is known that the product of two numbers is zero if and only if at least one of the numbers is zero. From the above it follows that x 1 − x 2 = 0 and/or x 1 + x 2 = 0, which is the same x 2 = x 1 and/or x 2 = − x 1. An obvious contradiction arose, because at first it was agreed that the root of the equation x 2 differs from x 1 And − x 1. So, we have proven that the equation has no roots other than x = - c a and x = - - c a.

Let us summarize all the arguments above.

Definition 6

Incomplete quadratic equation a x 2 + c = 0 is equivalent to the equation x 2 = - c a, which:

• will have no roots at - c a< 0 ;
• will have two roots x = - c a and x = - - c a for - c a > 0.

Let us give examples of solving the equations a x 2 + c = 0.

Example 3

Given a quadratic equation 9 x 2 + 7 = 0. It is necessary to find a solution.

Solution

Let's move the free term to the right side of the equation, then the equation will take the form 9 x 2 = − 7.
Let us divide both sides of the resulting equation by 9 , we arrive at x 2 = - 7 9 . On the right side we see a number with a minus sign, which means: the given equation has no roots. Then the original incomplete quadratic equation 9 x 2 + 7 = 0 will have no roots.

Answer: the equation 9 x 2 + 7 = 0 has no roots.

Example 4

The equation needs to be solved − x 2 + 36 = 0.

Solution

Let's move 36 to the right side: − x 2 = − 36.
Let's divide both parts by − 1 , we get x 2 = 36. On the right side there is a positive number, from which we can conclude that x = 36 or x = - 36 .
Let's extract the root and write down the final result: incomplete quadratic equation − x 2 + 36 = 0 has two roots x=6 or x = − 6.

Answer: x=6 or x = − 6.

Solution of the equation a x 2 +b x=0

Let us analyze the third type of incomplete quadratic equations, when c = 0. To find a solution to an incomplete quadratic equation a x 2 + b x = 0, we will use the factorization method. Let's factorize the polynomial that is on the left side of the equation, taking the common factor out of brackets x. This step will make it possible to transform the original incomplete quadratic equation into its equivalent x (a x + b) = 0. And this equation, in turn, is equivalent to a set of equations x = 0 And a x + b = 0. The equation a x + b = 0 linear, and its root: x = − b a.

Definition 7

Thus, the incomplete quadratic equation a x 2 + b x = 0 will have two roots x = 0 And x = − b a.

Let's reinforce the material with an example.

Example 5

It is necessary to find a solution to the equation 2 3 · x 2 - 2 2 7 · x = 0.

Solution

We'll take it out x outside the brackets we get the equation x · 2 3 · x - 2 2 7 = 0 . This equation is equivalent to the equations x = 0 and 2 3 x - 2 2 7 = 0. Now you should solve the resulting linear equation: 2 3 · x = 2 2 7, x = 2 2 7 2 3.

Briefly write the solution to the equation as follows:

2 3 x 2 - 2 2 7 x = 0 x 2 3 x - 2 2 7 = 0

x = 0 or 2 3 x - 2 2 7 = 0

x = 0 or x = 3 3 7

Answer: x = 0, x = 3 3 7.

Discriminant, formula for the roots of a quadratic equation

To find solutions to quadratic equations, there is a root formula:

Definition 8

x = - b ± D 2 · a, where D = b 2 − 4 a c– the so-called discriminant of a quadratic equation.

Writing x = - b ± D 2 · a essentially means that x 1 = - b + D 2 · a, x 2 = - b - D 2 · a.

It would be useful to understand how this formula was derived and how to apply it.

Derivation of the formula for the roots of a quadratic equation

Let us be faced with the task of solving a quadratic equation a x 2 + b x + c = 0. Let us carry out a number of equivalent transformations:

• divide both sides of the equation by a number a, different from zero, we obtain the following quadratic equation: x 2 + b a · x + c a = 0 ;
• Let's select the complete square on the left side of the resulting equation:
  x 2 + b a · x + c a = x 2 + 2 · b 2 · a · x + b 2 · a 2 - b 2 · a 2 + c a = = x + b 2 · a 2 - b 2 · a 2 + c a
  After this, the equation will take the form: x + b 2 · a 2 - b 2 · a 2 + c a = 0;
• Now it is possible to transfer the last two terms to the right side, changing the sign to the opposite, after which we get: x + b 2 · a 2 = b 2 · a 2 - c a ;
• Finally, we transform the expression written on the right side of the last equality:
  b 2 · a 2 - c a = b 2 4 · a 2 - c a = b 2 4 · a 2 - 4 · a · c 4 · a 2 = b 2 - 4 · a · c 4 · a 2 .

Thus, we arrive at the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2 , equivalent to the original equation a x 2 + b x + c = 0.

We analyzed the solution of similar equations in previous paragraphs(solving incomplete quadratic equations). The experience already gained makes it possible to draw a conclusion regarding the roots of the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2:

• with b 2 - 4 a c 4 a 2< 0 уравнение не имеет действительных решений;
• when b 2 - 4 · a · c 4 · a 2 = 0 the equation is x + b 2 · a 2 = 0, then x + b 2 · a = 0.

From here the only root x = - b 2 · a is obvious;

• for b 2 - 4 · a · c 4 · a 2 > 0, the following will be true: x + b 2 · a = b 2 - 4 · a · c 4 · a 2 or x = b 2 · a - b 2 - 4 · a · c 4 · a 2 , which is the same as x + - b 2 · a = b 2 - 4 · a · c 4 · a 2 or x = - b 2 · a - b 2 - 4 · a · c 4 · a 2 , i.e. the equation has two roots.

It is possible to conclude that the presence or absence of roots of the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2 (and therefore the original equation) depends on the sign of the expression b 2 - 4 · a · c 4 · a 2 written on the right side. And the sign of this expression is given by the sign of the numerator, (denominator 4 a 2 will always be positive), that is, the sign of the expression b 2 − 4 a c. This expression b 2 − 4 a c the name is given - the discriminant of the quadratic equation and the letter D is defined as its designation. Here you can write down the essence of the discriminant - based on its value and sign, they can conclude whether the quadratic equation will have real roots, and, if so, what is the number of roots - one or two.

Let's return to the equation x + b 2 · a 2 = b 2 - 4 · a · c 4 · a 2 . Let's rewrite it using discriminant notation: x + b 2 · a 2 = D 4 · a 2 .

Let us formulate our conclusions again:

Definition 9

• at D< 0 the equation has no real roots;
• at D=0 the equation has a single root x = - b 2 · a ;
• at D > 0 the equation has two roots: x = - b 2 · a + D 4 · a 2 or x = - b 2 · a - D 4 · a 2. Based on the properties of radicals, these roots can be written in the form: x = - b 2 · a + D 2 · a or - b 2 · a - D 2 · a. And, when we open the modules and bring the fractions to a common denominator, we get: x = - b + D 2 · a, x = - b - D 2 · a.

So, the result of our reasoning was the derivation of the formula for the roots of a quadratic equation:

x = - b + D 2 a, x = - b - D 2 a, discriminant D calculated by the formula D = b 2 − 4 a c.

These formulas make it possible to determine both real roots when the discriminant is greater than zero. When the discriminant is zero, applying both formulas will give the same root as the only solution to the quadratic equation. In the case where the discriminant is negative, if we try to use the quadratic root formula, we will be faced with the need to take the square root of a negative number, which will take us beyond the scope of real numbers. With a negative discriminant, the quadratic equation will not have real roots, but a pair of complex conjugate roots is possible, determined by the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

It is possible to solve a quadratic equation by immediately using the root formula, but this is generally done when it is necessary to find complex roots.

In the majority of cases, it usually means searching not for complex, but for real roots of a quadratic equation. Then it is optimal, before using the formulas for the roots of a quadratic equation, to first determine the discriminant and make sure that it is not negative (otherwise we will conclude that the equation has no real roots), and then proceed to calculate the value of the roots.

The reasoning above makes it possible to formulate an algorithm for solving a quadratic equation.

Definition 10

To solve a quadratic equation a x 2 + b x + c = 0, necessary:

• according to the formula D = b 2 − 4 a c find the discriminant value;
• at D< 0 сделать вывод об отсутствии у квадратного уравнения действительных корней;
• for D = 0, find the only root of the equation using the formula x = - b 2 · a ;
• for D > 0, determine two real roots of the quadratic equation using the formula x = - b ± D 2 · a.

Note that when the discriminant is zero, you can use the formula x = - b ± D 2 · a, it will give the same result as the formula x = - b 2 · a.

Let's look at examples.

Examples of solving quadratic equations

Let us give a solution to the examples for different meanings discriminant.

Example 6

We need to find the roots of the equation x 2 + 2 x − 6 = 0.

Solution

Let's write down the numerical coefficients of the quadratic equation: a = 1, b = 2 and c = − 6. Next we proceed according to the algorithm, i.e. Let's start calculating the discriminant, for which we will substitute the coefficients a, b And c into the discriminant formula: D = b 2 − 4 · a · c = 2 2 − 4 · 1 · (− 6) = 4 + 24 = 28 .

So we get D > 0, which means that the original equation will have two real roots.
To find them, we use the root formula x = - b ± D 2 · a and, substituting the corresponding values, we get: x = - 2 ± 28 2 · 1. Let us simplify the resulting expression by taking the factor out of the root sign and then reducing the fraction:

x = - 2 ± 2 7 2

x = - 2 + 2 7 2 or x = - 2 - 2 7 2

x = - 1 + 7 or x = - 1 - 7

Answer: x = - 1 + 7 ​​​​​​, x = - 1 - 7 .

Example 7

Need to solve a quadratic equation − 4 x 2 + 28 x − 49 = 0.

Solution

Let's define the discriminant: D = 28 2 − 4 · (− 4) · (− 49) = 784 − 784 = 0. With this value of the discriminant, the original equation will have only one root, determined by the formula x = - b 2 · a.

x = - 28 2 (- 4) x = 3.5

Answer: x = 3.5.

Example 8

The equation needs to be solved 5 y 2 + 6 y + 2 = 0

Solution

The numerical coefficients of this equation will be: a = 5, b = 6 and c = 2. We use these values ​​to find the discriminant: D = b 2 − 4 · a · c = 6 2 − 4 · 5 · 2 = 36 − 40 = − 4 . The calculated discriminant is negative, so the original quadratic equation has no real roots.

In the case when the task is to indicate complex roots, we apply the root formula, performing actions with complex numbers:

x = - 6 ± - 4 2 5,

x = - 6 + 2 i 10 or x = - 6 - 2 i 10,

x = - 3 5 + 1 5 · i or x = - 3 5 - 1 5 · i.

Answer: there are no real roots; the complex roots are as follows: - 3 5 + 1 5 · i, - 3 5 - 1 5 · i.

IN school curriculum There is no standard requirement to look for complex roots, therefore, if during the solution the discriminant is determined to be negative, the answer is immediately written down that there are no real roots.

Root formula for even second coefficients

The root formula x = - b ± D 2 · a (D = b 2 − 4 · a · c) makes it possible to obtain another formula, more compact, allowing one to find solutions to quadratic equations with an even coefficient for x (or with a coefficient of the form 2 · n, for example, 2 3 or 14 ln 5 = 2 7 ln 5). Let us show how this formula is derived.

Let us be faced with the task of finding a solution to the quadratic equation a · x 2 + 2 · n · x + c = 0 . We proceed according to the algorithm: we determine the discriminant D = (2 n) 2 − 4 a c = 4 n 2 − 4 a c = 4 (n 2 − a c), and then use the root formula:

x = - 2 n ± D 2 a, x = - 2 n ± 4 n 2 - a c 2 a, x = - 2 n ± 2 n 2 - a c 2 a, x = - n ± n 2 - a · c a .

Let the expression n 2 − a · c be denoted as D 1 (sometimes it is denoted D "). Then the formula for the roots of the quadratic equation under consideration with the second coefficient 2 · n will take the form:

x = - n ± D 1 a, where D 1 = n 2 − a · c.

It is easy to see that D = 4 · D 1, or D 1 = D 4. In other words, D 1 is a quarter of the discriminant. Obviously, the sign of D 1 is the same as the sign of D, which means the sign of D 1 can also serve as an indicator of the presence or absence of roots of a quadratic equation.

Definition 11

Thus, to find a solution to a quadratic equation with a second coefficient of 2 n, it is necessary:

• find D 1 = n 2 − a · c ;
• at D 1< 0 сделать вывод, что действительных корней нет;
• when D 1 = 0, determine the only root of the equation using the formula x = - n a;
• for D 1 > 0, determine two real roots using the formula x = - n ± D 1 a.

Example 9

It is necessary to solve the quadratic equation 5 x 2 − 6 x − 32 = 0.

Solution

We can represent the second coefficient of the given equation as 2 · (− 3) . Then we rewrite the given quadratic equation as 5 x 2 + 2 (− 3) x − 32 = 0, where a = 5, n = − 3 and c = − 32.

Let's calculate the fourth part of the discriminant: D 1 = n 2 − a · c = (− 3) 2 − 5 · (− 32) = 9 + 160 = 169. The resulting value is positive, which means that the equation has two real roots. Let us determine them using the corresponding root formula:

x = - n ± D 1 a, x = - - 3 ± 169 5, x = 3 ± 13 5,

x = 3 + 13 5 or x = 3 - 13 5

x = 3 1 5 or x = - 2

It would be possible to carry out calculations using the usual formula for the roots of a quadratic equation, but in this case the solution would be more cumbersome.

Answer: x = 3 1 5 or x = - 2 .

Simplifying the form of quadratic equations

Sometimes it is possible to optimize the form of the original equation, which will simplify the process of calculating the roots.

For example, the quadratic equation 12 x 2 − 4 x − 7 = 0 is clearly more convenient to solve than 1200 x 2 − 400 x − 700 = 0.

More often, simplification of the form of a quadratic equation is carried out by multiplying or dividing its both sides by a certain number. For example, above we showed a simplified representation of the equation 1200 x 2 − 400 x − 700 = 0, obtained by dividing both sides by 100.

Such a transformation is possible when the coefficients of the quadratic equation are not mutually prime numbers. Then we usually divide both sides of the equation by the largest common divisor absolute values its coefficients.

As an example, we use the quadratic equation 12 x 2 − 42 x + 48 = 0. Let us determine the GCD of the absolute values ​​of its coefficients: GCD (12, 42, 48) = GCD(GCD (12, 42), 48) = GCD (6, 48) = 6. Let us divide both sides of the original quadratic equation by 6 and obtain the equivalent quadratic equation 2 x 2 − 7 x + 8 = 0.

By multiplying both sides of a quadratic equation, you usually get rid of fractional coefficients. In this case, they multiply by the least common multiple of the denominators of its coefficients. For example, if each part of the quadratic equation 1 6 x 2 + 2 3 x - 3 = 0 is multiplied with LCM (6, 3, 1) = 6, then it will become written in more in simple form x 2 + 4 x − 18 = 0 .

Finally, we note that we almost always get rid of the minus at the first coefficient of a quadratic equation by changing the signs of each term of the equation, which is achieved by multiplying (or dividing) both sides by − 1. For example, from the quadratic equation − 2 x 2 − 3 x + 7 = 0, you can go to its simplified version 2 x 2 + 3 x − 7 = 0.

Relationship between roots and coefficients

The formula for the roots of quadratic equations, already known to us, x = - b ± D 2 · a, expresses the roots of the equation through its numerical coefficients. Based on this formula, we have the opportunity to specify other dependencies between the roots and coefficients.

The most famous and applicable formulas are Vieta’s theorem:

x 1 + x 2 = - b a and x 2 = c a.

In particular, for the given quadratic equation the sum of the roots is the second coefficient with opposite sign, and the product of the roots is equal to the free term. For example, by looking at the form of the quadratic equation 3 x 2 − 7 x + 22 = 0, it is possible to immediately determine that the sum of its roots is 7 3 and the product of the roots is 22 3.

You can also find a number of other connections between the roots and coefficients of a quadratic equation. For example, the sum of the squares of the roots of a quadratic equation can be expressed in terms of coefficients:

x 1 2 + x 2 2 = (x 1 + x 2) 2 - 2 x 1 x 2 = - b a 2 - 2 c a = b 2 a 2 - 2 c a = b 2 - 2 a c a 2.

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Let's consider the problem. Rectangle base more height by 10 cm, and its area is 24 cm². Find the height of the rectangle. Let X centimeters is the height of the rectangle, then its base is equal to ( X+10) cm. The area of ​​this rectangle is X(X+ 10) cm². According to the conditions of the problem X(X+ 10) = 24. Opening the brackets and moving the number 24 with the opposite sign to the left side of the equation, we get: X² + 10 X-24 = 0. When solving this problem, an equation was obtained that is called quadratic.

A quadratic equation is an equation of the form

ax ²+ bx+c= 0

Where a, b, c- given numbers, and A≠ 0, and X- unknown.

Odds a, b, c The quadratic equation is usually called: a— the first or highest coefficient, b- second coefficient, c- a free member. For example, in our problem, the leading coefficient is 1, the second coefficient is 10, and the free term is -24. Solving many problems in mathematics and physics comes down to solving quadratic equations.

Solving Quadratic Equations

Complete quadratic equations. The first step is to bring the given equation to standard form ax²+ bx+ c = 0. Let's return to our problem, in which the equation can be written as X(X+ 10) = 24 let’s bring it to standard form, open the brackets X² + 10 X- 24 = 0, we solve this equation using the formula for the roots of a general quadratic equation.

The expression under the root sign in this formula is called the discriminant D = b² - 4 ac

If D>0, then the quadratic equation has two different roots, which can be found using the formula for the roots of a quadratic equation.

If D=0, then the quadratic equation has one root.

If D<0, то квадратное уравнение не имеет действительных корней, т. е. не имеет решения.

Let's substitute the values ​​into our formula A= 1, b= 10, c= -24.

we get D>0, therefore we get two roots.

Let's consider an example where D=0, under this condition there should be one root.

25x² — 30 x+ 9 = 0

Consider an example where D<0, при этом условии решения не должно быть.

2x² + 3 x+ 4 = 0

The number under the root sign (discriminant) is negative; we write the answer as follows: the equation has no real roots.

Solving incomplete quadratic equations

Quadratic equation ax² + bx+ c= 0 is called incomplete if at least one of the coefficients b or c equal to zero. An incomplete quadratic equation is an equation of one of the following types:

ax² = 0,

ax² + c= 0, c≠ 0,

ax² + bx= 0, b≠ 0.

Let's look at a few examples and solve the equation

Dividing both sides of the equation by 5 gives the equation X² = 0, the answer will have one root X= 0.

Consider an equation of the form

3X² - 27 = 0

Dividing both sides by 3, we get the equation X² - 9 = 0, or it can be written X² = 9, the answer will have two roots X= 3 and X= -3.

Consider an equation of the form

2X² + 7 = 0

Dividing both sides by 2, we get the equation X² = -7/2. This equation has no real roots, since X² ≥ 0 for any real number X.

Consider an equation of the form

3X² + 5 X= 0

Factoring the left side of the equation, we get X(3X+ 5) = 0, the answer will have two roots X= 0, X=-5/3.

The most important thing when solving quadratic equations is to bring the quadratic equation to a standard form, memorize the formula for the roots of a general quadratic equation and not get confused in the signs.