Solve quadratic equations 8. Solve quadratic equations. Methods for solving complete quadratic equations

The lesson will introduce the concept of a quadratic equation, consider its two types: complete and incomplete. Special attention in the lesson will be paid to varieties of incomplete quadratic equations, many examples will be covered in the second half of the lesson.

Subject:Quadratic equations.

Lesson:Quadratic equations. Basic concepts

Definition.quadratic equation is called an equation of the form

Fixed real numbers, which define a quadratic equation. These numbers have specific names:

Senior coefficient (multiplier at );

Second coefficient (multiplier at );

Free member (number without multiplier-variable).

Comment. It should be understood that the indicated sequence of writing the terms in a quadratic equation is standard, but not mandatory, and in the case of their rearrangement, it is necessary to be able to determine the numerical coefficients not by their ordinal arrangement, but by belonging to the variables.

Definition. The expression is called square trinomial.

Example 1 Given a quadratic equation . Its odds are:

senior coefficient;

Second coefficient (note that the coefficient is indicated with a leading sign);

Free member.

Definition. If , then the quadratic equation is called unreduced, and if , then the quadratic equation is called given.

Example 2 Give a quadratic equation . Let's divide both parts by 2: .

Comment. As can be seen from the previous example, by dividing by the leading coefficient, we did not change the equation, but changed its form (made it reduced), similarly, it could also be multiplied by some non-zero number. Thus, the quadratic equation is not given by a single triplet of numbers, but it is said that is specified up to a nonzero set of coefficients.

Definition.Reduced quadratic equation is obtained from the unreduced by dividing by the leading factor , and it has the form:

.

The following designations are accepted: . Then reduced quadratic equation looks like:

.

Comment. In the above form of the quadratic equation, it can be seen that the quadratic equation can be specified with just two numbers: .

Example 2 (continued). Let us indicate the coefficients that define the reduced quadratic equation . , . These coefficients are also indicated taking into account the sign. The same two numbers define the corresponding unreduced quadratic equation .

Comment. The corresponding unreduced and reduced quadratic equations are the same, i.e. have the same set of roots.

Definition. Some of the coefficients in the unreduced form or in the reduced form of the quadratic equation may be zero. In this case, the quadratic equation is called incomplete. If all coefficients are non-zero, then the quadratic equation is called complete.

There are several types of incomplete quadratic equation.

If we have not yet considered the solution of the complete quadratic equation, then we can easily solve the incomplete one using the methods already known to us.

Definition.Solve a quadratic equation- means to find all values ​​of the variable (the roots of the equation), at which the given equation turns into the correct numerical equality, or to establish that there are no such values.

Example 3 Consider an example of this type of incomplete quadratic equations. Solve the equation.

Solution. Let's take out the common factor . We can solve equations of this type according to the following principle: the product is equal to zero if and only if one of the factors zero, and another one exists for this value of the variable. Thus:

Answer.; .

Example 4 Solve the equation.

Solution. 1 way. Factor it using the difference of squares formula

, therefore, similarly to the previous example or .

2 way. Move the free term to the right and extract Square root from both parts.

Answer. .

Example 5 Solve the equation.

Solution. We move the free term to the right, but , i.e. not in the equation a negative number equates to negative, which makes no sense for any values ​​of the variable, therefore, there are no roots.

Answer. There are no roots.

Example 6.Solve the equation.

Solution. Divide both sides of the equation by 7: .

Answer. 0.

Consider examples in which you first need to bring the quadratic equation to the standard form, and then solve it.

Example 7. Solve the equation.

Solution. To bring a quadratic equation to a standard form, it is necessary to transfer all the terms in one direction, for example, to the left, and bring similar ones.

An incomplete quadratic equation has been obtained, which we already know how to solve, we get that or .

Answer. .

Example 8 (text problem). The product of two consecutive natural numbers is twice the square of the smaller number. Find these numbers.

Solution. Text tasks, as a rule, are solved according to the following algorithm.

1) Drawing up a mathematical model. At this stage, it is necessary to translate the text of the problem into the language of mathematical symbols (make an equation).

Let some first natural number denote unknown, then the next after it (consecutive numbers) will be . The smallest of these numbers is the number, we write the equation according to the condition of the problem:

, Where . Mathematical model drawn up.

Quadratic equations are studied in grade 8, so there is nothing complicated here. The ability to solve them is essential.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a , b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, we note that all quadratic equations can be divided into three classes:

1. Have no roots;
2. They have exactly one root;
3. They have two different roots.

This is an important difference between quadratic and linear equations, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac .

This formula must be known by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant, you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people think. Take a look at the examples and you will understand everything yourself:

Task. How many roots do quadratic equations have:

1. x 2 - 8x + 12 = 0;
2. 5x2 + 3x + 7 = 0;
3. x 2 − 6x + 9 = 0.

We write the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So, the discriminant is positive, so the equation has two different roots. We analyze the second equation in the same way:
a = 5; b = 3; c = 7;
D \u003d 3 2 - 4 5 7 \u003d 9 - 140 \u003d -131.

The discriminant is negative, there are no roots. The last equation remains:
a = 1; b = -6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is equal to zero - the root will be one.

Note that coefficients have been written out for each equation. Yes, it's long, yes, it's tedious - but you won't mix up the odds and don't make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you “fill your hand”, after a while you will no longer need to write out all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not so much.

The roots of a quadratic equation

Now let's move on to the solution. If the discriminant D > 0, the roots can be found using the formulas:

The basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

1. x 2 - 2x - 3 = 0;
2. 15 - 2x - x2 = 0;
3. x2 + 12x + 36 = 0.

First equation:
x 2 - 2x - 3 = 0 ⇒ a = 1; b = −2; c = -3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 (−1) 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and be able to count, there will be no problems. Most often, errors occur when negative coefficients are substituted into the formula. Here, again, the technique described above will help: look at the formula literally, paint each step - and get rid of mistakes very soon.

Incomplete quadratic equations

It happens that the quadratic equation is somewhat different from what is given in the definition. For example:

1. x2 + 9x = 0;
2. x2 − 16 = 0.

It is easy to see that one of the terms is missing in these equations. Such quadratic equations are even easier to solve than standard ones: they do not even need to calculate the discriminant. So let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b \u003d c \u003d 0. In this case, the equation takes the form ax 2 \u003d 0. Obviously, such an equation has a single root: x \u003d 0.

Let's consider other cases. Let b \u003d 0, then we get an incomplete quadratic equation of the form ax 2 + c \u003d 0. Let's slightly transform it:

Since the arithmetic square root exists only from a non-negative number, the last equality only makes sense when (−c / a ) ≥ 0. Conclusion:

1. If an incomplete quadratic equation of the form ax 2 + c = 0 satisfies the inequality (−c / a ) ≥ 0, there will be two roots. The formula is given above;
2. If (−c / a )< 0, корней нет.

As you can see, the discriminant was not required - there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c / a ) ≥ 0. It is enough to express the value of x 2 and see what is on the other side of the equal sign. If there positive number there will be two roots. If negative, there will be no roots at all.

Now let's deal with equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factorize the polynomial:

Taking the common factor out of the bracket

The product is equal to zero when at least one of the factors is equal to zero. This is where the roots come from. In conclusion, we will analyze several of these equations:

Task. Solve quadratic equations:

1. x2 − 7x = 0;
2. 5x2 + 30 = 0;
3. 4x2 − 9 = 0.

x 2 − 7x = 0 ⇒ x (x − 7) = 0 ⇒ x 1 = 0; x2 = −(−7)/1 = 7.

5x2 + 30 = 0 ⇒ 5x2 = -30 ⇒ x2 = -6. There are no roots, because the square cannot be equal to a negative number.

4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 \u003d -1.5.

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'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 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 work.

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

Class: 8

Consider the standard (studied in the school mathematics course) and non-standard methods for solving quadratic equations.

1. Decomposition of the left side of the quadratic equation into linear factors.

Consider examples:

3) x 2 + 10x - 24 = 0.

6(x 2 + x - x) = 0 | : 6

x 2 + x - x - \u003d 0;

x(x - ) + (x - ) = 0;

x(x - ) (x + ) = 0;

= ; – .

Answer: ; – .

For independent work:

Solve quadratic equations using the method of factoring the left side of a quadratic equation into linear factors.

a) x 2 - x \u003d 0; d) x 2 - 81 = 0; g) x 2 + 6x + 9 = 0;

b) x 2 + 2x \u003d 0; e) 4x 2 - = 0; h) x 2 + 4x + 3 = 0;

c) 3x 2 - 3x = 0; f) x 2 - 4x + 4 = 0; i) x 2 + 2x - 3 = 0.

a) 0; 1

b) -2; 0

c) 0; 1

2. The method of selection of a full square.

Consider examples:

For independent work.

Solve quadratic equations using the full square method.

3. Solution of quadratic equations by formula.

ax 2 + in + c \u003d 0, (a | 4a

4a 2 x 2 + 4ab + 4ac = 0;

2ax + 2ax 2v + in 2 - in 2 + 4ac \u003d 0;

2 \u003d in 2 - 4ac; =±;

Consider examples.

For independent work.

Solve quadratic equations using the formula x 1,2 =.

4. Solving quadratic equations using the Vieta theorem (direct and inverse)

x 2 + px + q = 0 - reduced quadratic equation

by Vieta's theorem.

If then the equation has two identical roots in sign and it depends on the coefficient.

If p, then .

If p, then .

For example:

If then the equation has two roots of different sign, and the larger root will be if p and will be if p.

For example:

For independent work.

Without solving the quadratic equation, use the inverse Vieta theorem to determine the signs of its roots:

a, b, j, l - various roots;

c, e, h – negative;

d, f, g, i, m – positive;

5. Solution of quadratic equations by the “transfer” method.

For independent work.

Solve quadratic equations using the "flip" method.

6. Solving quadratic equations using the properties of its coefficients.

I. ax 2 + bx + c = 0, where a 0

1) If a + b + c \u003d 0, then x 1 \u003d 1; x 2 =

Proof:

ax 2 + bx + c = 0 |: a

x 2 + x + = 0.

According to Vieta's theorem

By condition a + b + c = 0, then b = -a - c. Next, we get

It follows from this that x 1 =1; x 2 = . Q.E.D.

2) If a - b + c \u003d 0 (or b \u003d a + c), then x 1 \u003d - 1; x 2 \u003d -

Proof:

According to Vieta's theorem

By condition a - b + c \u003d 0, i.e. b = a + c. Next we get:

Therefore, x 1 \u003d - 1; x 2 \u003d -.

Consider examples.

1) 345 x 2 - 137 x - 208 = 0.

a + b + c \u003d 345 - 137 - 208 \u003d 0

x 1 = 1; x 2 ==

2) 132 x 2 - 247 x + 115 = 0.

a + b + c = 132 -247 -115 = 0.

x 1 = 1; x 2 ==

Answer: 1;

For independent work.

Using the properties of the coefficients of a quadratic equation, solve the equations

II. ax 2 + bx + c = 0, where a 0

x 1.2 = . Let b = 2k, i.e. even. Then we get

x 1.2 = = = =

Consider an example:

3x 2 - 14x + 16 = 0.

D 1 \u003d (-7) 2 - 3 16 \u003d 49 - 48 \u003d 1

x 1 = = 2; x 2 =

Answer: 2;

For independent work.

a) 4x 2 - 36x + 77 = 0

b) 15x 2 - 22x - 37 = 0

c) 4x 2 + 20x + 25 = 0

d) 9x 2 - 12x + 4 = 0

Answers:

III. x 2 + px + q = 0

x 1.2 = - ± 2 - q

Consider an example:

x 2 - 14x - 15 = 0

x 1.2 = 7 = 7

x 1 \u003d -1; x 2 = 15.

Answer: -1; 15.

For independent work.

a) x 2 - 8x - 9 \u003d 0

b) x 2 + 6x - 40 = 0

c) x 2 + 18x + 81 = 0

d) x 2 - 56x + 64 = 0

7. Solving a quadratic equation using graphs.

a) x 2 - 3x - 4 \u003d 0

Answer: -1; 4

b) x 2 - 2x + 1 = 0

c) x 2 - 2x + 5 = 0

Answer: no solution

For independent work.

Solve quadratic equations graphically:

8. Solving quadratic equations with a compass and straightedge.

ax2 + bx + c = 0,

x 2 + x + = 0.

x 1 and x 2 are roots.

Let A(0; 1), C(0;

According to the secant theorem:

OV · OD = OA · OS.

Therefore we have:

x 1 x 2 = 1 OS;

OS = x 1 x 2

K(; 0), where = -

F(0; ) = (0; ) = )

1) Construct the point S(-; ) - the center of the circle and the point A(0;1).

2) Draw a circle with radius R = SA/

3) The abscissas of the points of intersection of this circle with the x-axis are the roots of the original quadratic equation.

3 cases are possible:

1) R > SK (or R > ).

The circle intersects the x-axis at the point B(x 1; 0) and D(x 2; 0), where x 1 and x 2 are the roots of the quadratic equation ax 2 + bx + c = 0.

2) R = SK (or R = ).

The circle touches the x-axis in anguish B 1 (x 1; 0), where x 1 is the root of the quadratic equation

ax2 + bx + c = 0.

3) R< SK (или R < ).

The circle has no common points with the x-axis, i.e. there are no solutions.

1) x 2 - 2x - 3 = 0.

Center S(-; ), i.e.

x 0 = = - = 1,

y 0 = = = – 1.

(1; – 1) is the center of the circle.

Let's draw a circle (S; AS), where A(0; 1).

9. Solving quadratic equations using a nomogram

For the solution, four-digit mathematical tables of V.M. Bradys (Plate XXII, p. 83).

The nomogram allows, without solving the quadratic equation x 2 + px + q = 0, to determine the roots of the equation by its coefficients. For example:

5) z2 + 4z + 3 = 0.

Both roots are negative. Therefore, we will make a replacement: z 1 = - t. We get a new equation:

t 2 - 4t + 3 = 0.

t 1 \u003d 1; t2 = 3

z 1 \u003d - 1; z 2 \u003d - 3.

Answer: - 3; - 1

6) If the coefficients p and q are out of scale, then perform the substitution z \u003d k t and solve the equation using the nomogram: z 2 + pz + q \u003d 0.

k 2 t 2 + p kt + q = 0. |: k 2

k is taken with the expectation that inequalities take place:

For independent work.

y 2 + 6y - 16 = 0.

y 2 + 6y = 16, |+ 9

y 2 + 6y + 9 = 16 + 9

y 1 = 2, y 2 = -8.

Answer: -8; 2

For independent work.

Solve geometrically the equation y 2 - 6y - 16 = 0.