Formulate the definition of the root of a quadratic equation. The roots of a quadratic equation. Solving incomplete quadratic equations

Just. According to formulas and clear simple rules. At the first stage

we need to bring the given equation to standard form, i.e. to the view:

If the equation is already given to you in this form, you do not need to do the first stage. The most important thing is right

determine all coefficients a, b and c.

Formula for finding the roots of a quadratic equation.

The expression under the root sign is called discriminant . As you can see, to find x, we

use only a, b and c. Those. odds from quadratic equation. Just carefully insert

values a, b and c into this formula and count. Substitute with their signs!

For example, in the equation:

a =1; b = 3; c = -4.

Substitute the values ​​and write:

Example almost solved:

This is the answer.

The most common mistakes are confusion with the signs of values a, b and With. Rather, with substitution

negative values ​​into the formula for calculating the roots. Here the detailed formula saves

with specific numbers. If there are problems with calculations, do it!

Suppose we need to solve the following example:

Here a = -6; b = -5; c = -1

We paint everything in detail, carefully, without missing anything with all the signs and brackets:

Often quadratic equations look slightly different. For example, like this:

Now take note of the practical techniques that dramatically reduce the number of errors.

First reception. Don't be lazy before solving a quadratic equation bring it to standard form.

What does this mean?

Suppose, after any transformations, you get the following equation:

Do not rush to write the formula of the roots! You will almost certainly mix up the odds a, b and c.

Build the example correctly. First, x squared, then without a square, then a free member. Like this:

Get rid of the minus. How? We need to multiply the whole equation by -1. We get:

And now you can safely write down the formula for the roots, calculate the discriminant and complete the example.

Decide on your own. You should end up with roots 2 and -1.

Second reception. Check your roots! By Vieta's theorem.

To solve the given quadratic equations, i.e. if the coefficient

x2+bx+c=0,

thenx 1 x 2 =c

x1 +x2 =−b

For a complete quadratic equation in which a≠1:

x 2 +bx+c=0,

divide the whole equation by a:

→ →

where x 1 and x 2 - roots of the equation.

Reception third. If your equation has fractional coefficients, get rid of the fractions! Multiply

equation for a common denominator.

Conclusion. Practical Tips:

1. Before solving, we bring the quadratic equation to the standard form, build it right.

2. If there is a negative coefficient in front of the x in the square, we eliminate it by multiplying everything

equations for -1.

3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding

factor.

4. If x squared is pure, the coefficient for it is equal to one, the solution can be easily checked by

Video lesson 2: Solving quadratic equations

Lecture: Quadratic equations

The equation

The equation- this is a kind of equality, in the expressions of which there is a variable.

solve the equation- means to find such a number instead of a variable that will lead it to the correct equality.

An equation may have one solution, or several, or none at all.

To solve any equation, it should be simplified as much as possible to the form:

Linear: a*x = b;

Square: a*x 2 + b*x + c = 0.

That is, any equation before solving must be converted to a standard form.

Any equation can be solved in two ways: analytical and graphical.

On the graph, the solution to the equation is considered to be the points at which the graph intersects the x-axis.

Quadratic equations

An equation can be called quadratic if, when simplified, it takes the form:

a*x 2 + b*x + c = 0.

Wherein a, b, c are coefficients of the equation that differ from zero. BUT "X"- root of the equation. It is believed that a quadratic equation has two roots or may not have a solution at all. The resulting roots may be the same.

"a"- the coefficient that stands in front of the root in the square.

"b"- stands before the unknown in the first degree.

"With"- free term of the equation.

If, for example, we have an equation of the form:

2x 2 -5x+3=0

In it, "2" is the coefficient at the highest term of the equation, "-5" is the second coefficient, and "3" is the free term.

Solving a quadratic equation

There are many ways to solve a quadratic equation. However, in the school mathematics course, the solution is studied using the Vieta theorem, as well as using the discriminant.

Discriminant solution:

When solving using this method, it is necessary to calculate the discriminant using the formula:

If during the calculations you got that the discriminant is less than zero, this means that this equation has no solutions.

If the discriminant is zero, then the equation has two identical solutions. In this case, the polynomial can be collapsed according to the abbreviated multiplication formula into the square of the sum or difference. Then solve it like a linear equation. Or use the formula:

If the discriminant is greater than zero, then the following method must be used:

Vieta's theorem

If the equation is reduced, that is, the coefficient at the highest term is equal to one, then you can use Vieta's theorem.

So let's say the equation is:

The roots of the equation are found as follows:

Incomplete quadratic equation

There are several options for obtaining an incomplete quadratic equation, the form of which depends on the presence of coefficients.

1. If the second and third coefficients are equal to zero (b=0, c=0), then the quadratic equation will look like:

This equation will have a unique solution. Equality will only be true if the solution to the equation is zero.

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

Solving full quadratic equations is a bit more complicated (just a little bit) than those given.

Remember, any quadratic equation can be solved using the discriminant!

Even incomplete.

The rest of the 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 the discriminant.

Solving quadratic equations in this way is very simple, the main thing is to remember the sequence of actions and a couple of formulas.

If, then the equation has 2 roots. Pay special attention to step 2.

The discriminant D tells us the number of roots of the equation.

• If, then the formula at the step will be reduced to. Thus, the equation will have only 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 us turn to the geometric meaning of the quadratic equation.

The graph of the function is a parabola:

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

Example 9

Solve the Equation

Step 1 skip.

Step 2

Finding the discriminant:

So the equation has two roots.

Step 3

Answer:

Example 10

Solve the Equation

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

Step 2

Finding the discriminant:

So the equation has one root.

Answer:

Example 11

Solve the Equation

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

Step 2

Finding the discriminant:

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

Now we know how to write down such answers correctly.

Answer: no roots

2. Solving quadratic equations using the Vieta theorem

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

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

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

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.

Example 12

Solve the Equation

This equation is suitable for solution using Vieta's theorem, because .

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

And the product is:

Let's create and solve the system:

• and. The sum is;
• and. The sum is;
• and. The amount is equal.

and are the solution of the system:

Answer: ; .

Example 13

Solve the Equation

Answer:

Example 14

Solve the Equation

The equation is reduced, 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 - unknown, - some numbers, moreover.

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

Because if, the equation will immediately become 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 - complete.

Methods for solving incomplete quadratic equations

To begin with, we will analyze the methods for solving incomplete quadratic equations - they are simpler.

The following types of equations can be distinguished:

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 consider the solution of each of these subtypes.

Obviously, this equation always has only one root:

A number squared cannot be negative, because when multiplying 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

These formulas do not need to be memorized. The main thing to remember is that it cannot be less.

Examples of solving quadratic equations

Example 15

Answer:

Never forget about roots with a negative sign!

Example 16

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

no roots.

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

Answer:

Example 17

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:

We factorize the left side of the equation and find the roots:

Answer:

Methods for solving complete quadratic equations

1. Discriminant

Solving quadratic equations in 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 the discriminant! Even incomplete.

Did you notice the root of the discriminant in the root formula?

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 a root:
• If, then the equation has the same root, but 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 are there different numbers of roots?

Let us turn to the geometric meaning of the quadratic equation. The graph of the function is a parabola:

In a particular case, which is a quadratic equation, .

And this means that the roots of the quadratic equation are the points of intersection with the x-axis (axis).

The parabola may not cross the axis at all, or it may intersect it at one (when the top 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 upwards, and if - then downwards.

4 examples of solving quadratic equations

Example 18

Answer:

Example 19

Answer: .

Example 20

Answer:

Example 21

This means there are no solutions.

Answer: .

2. Vieta's theorem

Using Vieta's theorem is very easy.

All you need is pick up such a pair of numbers, the product of which 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 to given quadratic equations ().

Let's look at a few examples:

Example 22

Solve the equation.

Solution:

This equation is suitable for solution using Vieta's theorem, because . Other coefficients: ; .

The sum of the roots of the equation is:

And the product is:

Let's select such pairs of numbers, the product of which is equal, and check if their sum is equal:

• and. The sum is;
• and. The sum is;
• and. The amount is equal.

and are the solution of the system:

Thus, and are the roots of our equation.

Answer: ; .

Example 23

Solution:

We select such pairs of numbers that give in the product, and then check whether their sum is equal:

and: give in total.

and: give in total. To get it, you just need to change the signs of the alleged roots: and, after all, the product.

Answer:

Example 24

Solution:

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

We select such pairs of numbers that give in the product, and the difference of which is equal to:

and: their difference is - not suitable;

and: - not suitable;

and: - not suitable;

and: - suitable. It remains only to remember that one of the roots is negative. Since their sum must be equal, then the root, which is smaller in absolute value, must be negative: . We check:

Answer:

Example 25

Solve the equation.

Solution:

The equation is reduced, which means:

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

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

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

Answer:

Example 26

Solve the equation.

Solution:

The equation is reduced, 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 are minus.

We select such pairs of numbers, the product of which is equal to:

Obviously, the roots are the numbers and.

Answer:

Agree, it is very convenient - to invent roots orally, instead of counting this nasty discriminant.

Try to use Vieta's theorem as often as possible!

But the Vieta theorem is needed in order to facilitate and speed up finding the roots.

To make it profitable for you to use it, you must bring the actions to automatism. And for this, solve five more examples.

But don't cheat: you can't use the discriminant! Only Vieta's theorem!

5 examples of Vieta's theorem for self-study

Example 27

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

According to Vieta's theorem:

As usual, we start the selection with the product:

Not suitable because the amount;

: the amount is what you need.

Answer: ; .

Example 28

Task 2.

And again, our favorite Vieta theorem: the sum should work out, but the product is equal.

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

Answer: ; .

Example 29

Task 3.

Hmm... Where is it?

It is necessary to transfer all the terms into one part:

The sum of the roots is equal to the product.

Yes, stop! The equation is not given.

But Vieta's theorem is applicable only in the given equations.

So first you need to bring the equation.

If you can’t bring it up, drop this idea and solve it in another way (for example, through the discriminant).

Let me remind you that to bring a quadratic equation means to make the leading coefficient equal to:

Then the sum of the roots is equal, and the product.

It's easier to pick up here: after all - a prime number (sorry for the tautology).

Answer: ; .

Example 30

Task 4.

The free term is negative.

What's so special about it?

And the fact that the roots will be of different signs.

And now, during the selection, we check not the sum of the roots, but the difference between their modules: this difference is equal, but the product.

So, the roots are equal and, but one of them is with a 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: ; .

Example 31

Task 5.

What needs to be done 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 and, but one of them is minus. Which? Their sum must be equal, which means that with a minus there will be a larger root.

Answer: ; .

Summarize

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

3. Full square selection method

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

For example:

Example 32

Solve the equation: .

Solution:

Answer:

Example 33

Solve the equation: .

Solution:

Answer:

In general, the transformation will look like this:

This implies: .

Doesn't it remind you of anything?

It's the discriminant! That's exactly how the discriminant formula was obtained.

QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN

Quadratic equation is an equation of the form, where is the unknown, are the coefficients of the quadratic equation, is 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 free term c are equal to zero:

• if the coefficient, the equation has the form: ,
• if a free term, the equation has the form: ,
• if and, the equation has the form: .

1. Algorithm for solving incomplete quadratic equations

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

1) 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 the discriminant

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

2) 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 a root, 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 (an equation of the form, where) is equal, and the product of the roots is equal, i.e. , a.

2.3. Full square solution

Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factorization of a square trinomial. Geometric interpretation. Examples of determining roots and factorization.

Content

See also: Solving quadratic equations online

Basic Formulas

Consider the quadratic equation:
(1) .
The roots of a quadratic equation(1) are determined by the formulas:
; .
These formulas can be combined like this:
.
When the roots of the quadratic equation are known, then the polynomial of the second degree can be represented as a product of factors (factored):
.

Further, we assume that are real numbers.
Consider discriminant of a quadratic equation:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
; .
Then the factorization of the square trinomial has the form:
.
If the discriminant is zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is the imaginary unit, ;
and are the real and imaginary parts of the roots:
; .
Then

.

Graphic interpretation

If build function graph
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
When , the graph crosses the abscissa axis (axis) at two points ().
When , the graph touches the x-axis at one point ().
When , the graph does not cross the x-axis ().

Useful Formulas Related to Quadratic Equation

(f.1) ;
(f.2) ;
(f.3) .

Derivation of the formula for the roots of a quadratic equation

We perform transformations and apply formulas (f.1) and (f.3):




,
where
; .

So, we got the formula for the polynomial of the second degree in the form:
.
From this it can be seen that the equation

performed at
and .
That is, and are the roots of the quadratic equation
.

Examples of determining the roots of a quadratic equation

Example 1

(1.1) .

.
Comparing with our equation (1.1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.

From here we obtain the decomposition of the square trinomial into factors:

.

Graph of the function y = 2 x 2 + 7 x + 3 crosses the x-axis at two points.

Let's plot the function
.
The graph of this function is a parabola. It crosses the x-axis (axis) at two points:
and .
These points are the roots of the original equation (1.1).

;
;
.

Example 2

Find the roots of a quadratic equation:
(2.1) .

We write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.

Then the factorization of the trinomial has the form:
.

Graph of the function y = x 2 - 4 x + 4 touches the x-axis at one point.

Let's plot the function
.
The graph of this function is a parabola. It touches the x-axis (axis) at one point:
.
This point is the root of the original equation (2.1). Since this root is factored twice:
,
then such a root is called a multiple. That is, they consider that there are two equal roots:
.

;
.

Example 3

Find the roots of a quadratic equation:
(3.1) .

We write the quadratic equation in general form:
(1) .
Let us rewrite the original equation (3.1):
.
Comparing with (1), we find the values ​​of the coefficients:
.
Finding the discriminant:
.
The discriminant is negative, . Therefore, there are no real roots.

You can find complex roots:
;
;
.

Then


.

The graph of the function does not cross the x-axis. There are no real roots.

Let's plot the function
.
The graph of this function is a parabola. It does not cross the abscissa (axis). Therefore, there are no real roots.

There are no real roots. Complex roots:
;
;
.

See also:

”, that is, equations of the first degree. In this lesson, we will explore what is a quadratic equation and how to solve it.

What is a quadratic equation

Important!

The degree of an equation is determined by the highest degree to which the unknown stands.

If the maximum degree to which the unknown stands is “2”, then you have a quadratic equation.

Examples of quadratic equations

• 5x2 - 14x + 17 = 0
• −x 2 + x +

  1

  3

  = 0
• x2 + 0.25x = 0
• x 2 − 8 = 0

Important! The general form of the quadratic equation looks like this:

A x 2 + b x + c = 0

"a", "b" and "c" - given numbers.

• "a" - the first or senior coefficient;
• "b" - the second coefficient;
• "c" is a free member.

To find "a", "b" and "c" You need to compare your equation with the general form of the quadratic equation "ax 2 + bx + c \u003d 0".

Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.

5x2 - 14x + 17 = 0 −7x 2 − 13x + 8 = 0 −x 2 + x +

The equation	Odds
a=5 b = −14 c = 17
a = −7 b = −13 c = 8

1

3

= 0

• a = -1
• b = 1
• c =

  1

  3

x2 + 0.25x = 0

• a = 1
• b = 0.25
• c = 0

x 2 − 8 = 0

• a = 1
• b = 0
• c = −8

How to solve quadratic equations

Unlike linear equations to solve quadratic equations, a special formula for finding roots.

Remember!

To solve a quadratic equation you need:

• bring the quadratic equation to general view"ax 2 + bx + c = 0". That is, only "0" should remain on the right side;
• use the formula for roots:

Let's use an example to figure out how to apply the formula to find the roots of a quadratic equation. Let's solve the quadratic equation.

X 2 - 3x - 4 = 0

The equation "x 2 - 3x - 4 = 0" has already been reduced to the general form "ax 2 + bx + c = 0" and does not require additional simplifications. To solve it, we need only apply formula for finding the roots of a quadratic equation.

Let's define the coefficients "a", "b" and "c" for this equation.

x 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =

With its help, any quadratic equation is solved.

In the formula "x 1; 2 \u003d" the root expression is often replaced
"b 2 − 4ac" to the letter "D" and called discriminant. The concept of a discriminant is discussed in more detail in the lesson "What is a discriminant".

Consider another example of a quadratic equation.

x 2 + 9 + x = 7x

In this form, it is rather difficult to determine the coefficients "a", "b", and "c". Let's first bring the equation to the general form "ax 2 + bx + c \u003d 0".

X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x2 + 9 - 6x = 0
x 2 − 6x + 9 = 0

Now you can use the formula for the roots.

X 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
x=

6

2

x=3
Answer: x = 3

There are times when there are no roots in quadratic equations. This situation occurs when a negative number appears in the formula under the root.