How to solve a quadratic equation using a discriminant. Properties of coefficients of a quadratic equation. Simplifying the form of quadratic equations

Quadratic equation, or algebraic equation of the 2nd degree with one unknown in general view is written as follows:

Ax 2 + bx + c = 0,

a, b, c are known coefficients, and a ≠ 0.
x is unknown.

3x 2 + 8x - 5 = 0.

2. Types of quadratic equations

Dividing both sides of the equation by a, we get reduced quadratic equation:

x 2 + px + q = 0,

p = b/a
q = c/a

If one of the coefficients b, c or both are equal to 0 at the same time, then a quadratic equation is called incomplete.

x 2 +8x-5=0 is a complete reduced quadratic equation.
3x 2 -5=0 is not a complete unreduced quadratic equation.
x 2 -8x=0 is not a complete reduced quadratic equation.

Incomplete quadratic equation of the form

X 2 = m

the simplest and most important, because the decision of every quadratic equation.

Three cases are possible:

m = 0, x = 0
m > 0, x = ±√‾m
m< 0, x = ±i√‾m. Где i — мнимая единица, равная √‾-1.

3. Solving a quadratic equation

The roots of an unreduced complete quadratic equation are found by the formula

x = (-b ± √‾(b 2 - 4ac)) / 2a

x = (7 ± √‾(1)) / 6

4. Properties of the roots of a quadratic equation. Discriminant.

According to the formula for the roots of a quadratic equation, there can be three cases, determined by the radical expression (b 2 - 4ac). It's called discriminant(discriminating).

Denoting the discriminant with the letter D, we can write:

D > 0, the equation has two different real roots.
D = 0, the equation has two equal real roots.
D< 0, уравнение имеет два различных мнимых корня.

x = (-b ± √‾(b 2 - 4ac)) / 2a

x = (7 ± √‾(7 2 - 4×3×4)) / (2×3)

x = (7 ± √‾(1)) / 6

5. Formulas useful in life

Often there are problems of converting volume into area or length and the inverse problem - converting area into volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be covered with boards contained in a certain volume, see.

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

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 2 roots. You need to pay special attention to step 2.

The discriminant D 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 turn to geometric sense quadratic equation.

The graph of the function is a parabola:

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.

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 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:

QUADRATE EQUATIONS. MIDDLE 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.

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.

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 free member 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 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 down that a 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:

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 us turn to the geometric meaning of the 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.

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 in reduced quadratic equations ().

Let's look at a few examples:

Example 22

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 23

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 24

Solution:

The free term of the equation is negative, and therefore the product of the roots is 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: - 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 25

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 26

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!

5 examples on Vieta’s theorem for independent work

Example 27

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: ; .

Example 28

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: ; .

Example 29

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:

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

It’s as easy as pie to choose here: after all, it’s a prime number (sorry for the tautology).

Answer: ; .

Example 30

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: ; .

Example 31

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's sum it up

Vieta's theorem is used only in the quadratic equations given.
Using Vieta's theorem, you can find the roots by selection, orally.
If the equation is not given or no suitable pair of factors of the free term is found, then 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 32

Solve the equation: .

Solution:

Answer:

Example 33

Solve the equation: .

Solution:

Answer:

In general, the transformation will look like this:

It follows: .

Doesn't remind you of anything?

This is a discriminatory thing! That's exactly how we got the discriminant formula.

QUADRATE 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

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

Content

See also: Solving quadratic equations online

Basic formulas

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

We further assume that - real numbers.
Let's 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 quadratic trinomial has the form:
.
If the discriminant is equal to 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 you build graph of a function
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
At , the graph intersects 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 carry out transformations and apply formulas (f.1) and (f.3):

,
Where
; .

So, we got the formula for a polynomial of the second degree in the form:
.
This shows 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:
.
We find the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.

From here we obtain the factorization of the quadratic trinomial:

.

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

Let's plot the function
.
The graph of this function is a parabola. It crosses the abscissa 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) .

Let's write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values of the coefficients:
.
We find 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). Because this root is factored twice:
,
then such a root is usually called a multiple. That is, they believe that there are two equal roots:
.

;
.

Example 3

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

Let's write the quadratic equation in general form:
(1) .
Let's rewrite the original equation (3.1):
.
Comparing with (1), we find the values of the coefficients:
.
We find 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 intersect the x-axis (axis). Therefore there are no real roots.

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