An algorithm for finding the roots of a quadratic equation is given. Let's compose an algorithm for solving a quadratic equation

1. Find the discriminant D according to the formula D= -4ac.

2.If D<0, то квадратное уравнение не имеет корней.

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

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

Now let's start solving our equation 3 -10x+3=0,

where =3, b=-10 and c=3.

Finding the discriminant:

D= -4*3*3=64

Since D>0, then this equation has two roots. We find them:

; .

Thus, the roots of the polynomial f(x)=3 -10+3 will be the numbers 3 and .

Horner's scheme

Horner's scheme(or Horner's rule, Horner's method) - an algorithm for calculating the value of a polynomial, written as a sum of polynomials (monomials), for a given value of a variable . She, in turn, helps us find out whether the number is the root of a given polynomial or not.

First, consider how the polynomial is divided f(x) into a binomial g(x).

This can be written as follows: f(x):g(x)=n(x), where f(x)- dividend, g(x)- divisor a n(x)- private.

But in the case when f(x) not divisible by g(x) there is a general notation of the expression

Here, the degree r(x)< deg s(x), в таком случае можно сказать, что делится на с остатком .

Consider dividing a polynomial by a binomial. Let be

,

We get

Where r is a number because the degree of r must be less than the degree of (x-c).

Let's multiply s(x) on and get

Thus, when dividing by a binomial, it is possible to determine the coefficients of the quotient from the obtained formulas. This method of determining the coefficients is called Horner's scheme.

				...
	+			...
c				...	r

Now let's look at a few examples of the application of Horner's scheme.

Example. Perform polynomial division f(x)= on the x+3.

Decision. At the beginning it is necessary to write x+3) as ( x-(-3)), since exactly -3 will participate in the scheme itself. In the top line we will write the coefficients, in the bottom line - the result of the actions.

f(x)=(x-2)(1)+16.

Finding roots according to Horner's scheme. Root types

According to Horner's scheme, one can find integer roots of a polynomial f(x). Let's look at this with an example.

Example. Find all integer roots of a polynomial f(x)= , using the Horner scheme.

Decision. The coefficients of this polynomial are integers. The coefficient before the highest degree (in our case before) is equal to one. Therefore, we will look for the integer roots of the polynomial among the divisors of the free term (we have 15), these are numbers:

Let's start with the number 1.

Table #1

			-21	-20
	+			-18	-38
			-18	-38

From the resulting table it can be seen that for =1 the polynomial of the polynomial f(x)= , we got the remainder r=192, not 0, which means that the unit is not a root. Therefore, we continue the check at =-1. To do this, we will not create a new table, but continue in the old one, and cross out the data that is no longer necessary.

Table number 2

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22

As we can see from the table, the last cell turned out to be zero, which means that r=0. Hence? the number -1 is the root of this polynomial. Dividing our polynomial polynomial f(x)= on ()=x+1 we got a polynomial

f(x)=(x+1)(),

the coefficients for which we took from the third line of table No. 2.

We can also make the equivalent notation

(x+1)(). Tag him (1)

Now it is necessary to continue the search for integer roots, but only now we will already look for the roots of the polynomial. We will look for these roots among the free term of the polynomial, the number 45.

Let's check the number -1 again.

Table #3

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22

Thus, the number -1 is the root of the polynomial, it can be written as

Taking into account equality (2), we can write equality (1) in the following form

Now we are looking for roots for the polynomial, again among the divisors of the free term. Let's check the number -1 again.

Table No. 4

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21

According to the table, we see that the number -1 is the root of the polynomial.

Given (3*), we can rewrite equality (2*) as:

Now we will look for the root for . Again we look at the divisors of the free term. Let's start checking again with the number -1.

Table number 5

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21
	+	-1
-1		-2	-19

We got a remainder that is not equal to zero, which means that the number -1 is not a root for the polynomial. Let's check the next number 1.

Table No. 6

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21
	+	-1
-1		-2	-19
	+			-21
			-21

And we see that again it does not fit, the remainder is r(x) = 24. We take a new number.

Let's check the number 3.

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21
	+	-1
-1		-2	-19
	+			-21
			-21
	+			-45
			-15

Table number 7

r(x)= 0, this means that the number 3 is the root of the polynomial, we can write this polynomial as:

=(x-3)( )

Given the resulting expression, we can write equality (5) as follows:

(x-3)( ) (6)

Let's check now for the polynomial

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21
	+	-1
-1		-2	-19
	+			-21
			-21
	+			-45
			-15
	+

Table No. 8

Based on the table, we see that the number 3 is the root of the polynomial . Now let's write the following:

We write the equality (5*), taking into account the resulting expression, as follows:

(x-3)()= = .

Find the root for the binomial among the divisors of the free term.

Let's take the number 5

Table No. 9

			-21	-20
	+			-18	-38
			-18	-38
	+	-1	-1		-2	-69	-45
-1			-22
	+	-1			-24	-45
-1			-22
	+	-1			-45
-1		-1	-21
	+	-1
-1		-2	-19
	+			-21
			-21
	+			-45
			-15
	+
	+	-5
-5

r(x)=0, so 5 is the root of the binomial.

Thus, we can write

Decision this example will be table number 8.

As can be seen from the table, the numbers -1; 3; 5 are the roots of the polynomial.

Now let's go directly to types of roots.

1 is the root of the third degree, since the bracket (x + 1) is in the third degree;

3- root of the second degree, bracket (x-3) in the second degree;

5 is the root of the first degree or, in other words, simple.

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In the term "quadratic equation" the key word is "quadratic". This means that the equation must necessarily contain a variable (the same X) in the square, and at the same time there should not be Xs in the third (or greater) degree.

The solution of many equations is reduced to the solution of quadratic equations.

Let's learn to determine that we have a quadratic equation, and not some other.

Example 1

Get rid of the denominator and multiply each term of the equation by

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

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

Example 2

Multiply the left and right sides by:

This equation, although it was originally in it, is not a square!

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

You see, it has shrunk - 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 conditionally 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 the complete quadratic equations, there are given 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 free term c are equal to zero:

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

Why did they come up with such a division? It would seem that there is an X squared, and okay. Such a division is due to the methods of solution. Let's consider each of them in more detail.

Solving incomplete quadratic equations

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

Incomplete quadratic equations are of types:

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. Since 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. These formulas do not need to be memorized. The main thing is that you should always know and remember that it cannot be less.

Let's try to solve some examples.

Example 5:

Solve the Equation

Now it remains to extract the root from the left and right parts. After all, do you remember how to extract the roots?

Answer:

Never forget about roots with a negative sign!!!

Example 6:

Solve the Equation

Answer:

Example 7:

Solve the Equation

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

no roots!

For such equations in which there are 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:

Here we will do without examples.

Solving complete quadratic equations

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

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 a root Special attention draw a step. The discriminant () 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 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. Solution of 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.

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. MIDDLE 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.

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

In this case, and can be equal to zero. In this stool 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:

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. So:

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:

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 that the 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 zero if at least one of the factors zero. This means that the equation has a solution when:

So, this quadratic equation has two roots: and.

Example:

Solve the equation.

Decision:

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.

Examples:

Solutions:

Answer:

Answer: .

Answer:

This means there are no solutions.

Answer: .

2. Vieta's theorem

Using the Vieta theorem is very easy: 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 to given quadratic equations ().

Let's look at a few examples:

Example #1:

Solve the equation.

Decision:

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 #2:

Decision:

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 #3:

Decision:

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 #4:

Solve the equation.

Decision:

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 #5:

Solve the equation.

Decision:

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:

Solutions for tasks for independent work:

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: ; .

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: ; .

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:

Fine. 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: ; .

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: ; .

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: ; .

Let me 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 1:

Solve the equation: .

Decision:

Answer:

Example 2:

Solve the equation: .

Decision:

Answer:

AT general view 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) We bring the equation to 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

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

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For successful passing the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who received a good education, earn much more than those who did not receive it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

Find a collection anywhere you want necessarily with solutions detailed analysis and decide, decide, decide!

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In conclusion...

If you don't like our tasks, find others. Just don't stop with theory.

“Understood” and “I know how to solve” are completely different skills. You need both.

Find problems and solve!

slide 2

Quadratic equations cycle of algebra lessons in the 8th grade according to the textbook by A.G. Mordkovich

Teacher MBOU Grushevskaya secondary school Kireeva T.A.

slide 3

Objectives: to introduce the concepts of a quadratic equation, the root of a quadratic equation; show solutions of quadratic equations; to form the ability to solve quadratic equations; show a way to solve complete quadratic equations using the formula of the roots of a quadratic equation.

slide 4

slide 5

A bit of history Quadratic equations in Ancient Babylon. The need to solve equations not only of the first, but also of the second degree, even in antiquity was caused by the need to solve problems related to finding the areas of land and with earthworks military nature, as well as with the development of astronomy and mathematics itself. The Babylonians knew how to solve quadratic equations about 2000 years before our faith. Applying modern algebraic notation, one can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations.

slide 6

The rule for solving these equations, set forth in the Babylonian texts, coincides with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions set out in the form of recipes, with no indication of how they were found. In spite of high level development of algebra in Babylonia, in cuneiform texts there is no concept of a negative number and general methods for solving quadratic equations.

Slide 7

Definition 1. A quadratic equation is an equation of the form where the coefficients a, b, c are any real numbers, and the Polynomial is called a square trinomial. a is the first or highest coefficient c is the second coefficient c is a free term

Slide 8

Definition 2. A quadratic equation is called reduced if its leading coefficient is equal to 1; a quadratic equation is called unreduced if the leading coefficient is different from 1. Example. 2 - 5 + 3 = 0 - unreduced quadratic equation - reduced quadratic equation

Slide 9

Definition 3. A complete quadratic equation is a quadratic equation in which all three terms are present. a + in + c \u003d 0 An incomplete quadratic equation is an equation in which not all three terms are present; is an equation for which at least one of the coefficients in, c is equal to zero.

Slide 10

Methods for solving incomplete quadratic equations.

slide 11

Solve tasks No. 24.16 (a, b) Solve the equation: or Answer. or Answer.

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Definition 4 The root of a quadratic equation is any value of the variable x at which the square trinomial vanishes; such a value of the variable x is also called the root of a square trinomial. Solving a quadratic equation means finding all its roots or establishing that there are no roots.

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The discriminant of a quadratic equation D 0 D=0 The equation has no roots The equation has two roots The equation has one root Formulas for the roots of a quadratic equation

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D>0 the quadratic equation has two roots, which are found by the formulas Example. Solve the equation Solution. a \u003d 3, b \u003d 8, c \u003d -11, Answer: 1; -3

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Algorithm for solving a quadratic equation 1. Calculate the discriminant D using the formula D = 2. If D 0, then the quadratic equation has two roots.

Programming in Lazarus for schoolchildren.

Lesson number 12.

Solution of a quadratic equation.

Matytsin Igor Vladimirovich

Teacher of mathematics and computer science

MBOU secondary school with. damsel

Purpose: to write a program for solving a quadratic equation, given any input.

Girl 2013.

The quadratic equation is one of the most common school course equations. Although it is quite easy to solve, sometimes you need to check the answers. For this you can use a simple program. It won't take long to write it.

You need to start with the quadratic equation itself. From the algebra course, we know that a quadratic equation is an equation of the form ax 2 + bx + c =0, where x - variable, a , b and c are some numbers, and a .

It can be seen from the definition that only the coefficients change in the equation a , b and c . These are the parameters we will enter into our program, and for this we will create three input fields from the components.

Fig 14.1 Input fields for coefficients.

It also follows from the definition that a . In this case, the equation will not be quadratic. And we will check this condition first of all. Let's create the "Solve" button and its event developer using the operator if check the condition a . And if a =0 we say that our equation is not quadratic.Here is the event handler for the button:procedure TForm1.Button1Click(Sender: TObject); var a,b,c:real; begin a:=strtofloat(edit1.Text); b:=strtofloat(edit2.Text); c:=strtofloat(edit3.Text); if a=0 then Label4.Caption:="The equation is not square";end;

Rice. 14.2 Testing for the existence of an equation.

Now it is necessary to describe what will happen if the equation is quadratic. This will also be in the same statement if after the word else and when using the compound operator.

If the equation is quadratic, then we will immediately solve it using the formula of the discriminant and the roots of the quadratic equation.

We find the discriminant by the formula: D := b * b – 4* a * c ;

If the discriminant is less than zero, then the equation has no solutions. It will be described like this:

If d then label 4. Caption :='Equation has no solutions' else …

And then else there will be a direct search for the roots of the equation using the formulas:

X1:=(-b+sqrt(D))/2*a;

X2:=(-b-sqrt(D))/2*a;

Here is the complete operator code if :

if a=0 then Label4.Caption:="The equation is not square" else

begin

D:=b*b-4*a*c;

if d

begin

X1:=(-b+sqrt(D))/2*a;

X2:=(-b-sqrt(D))/2*a;

Label4.Caption:="X1="+floattostr(x1)+" X2="+floattostr(x2);

end;

end;

Rice. 14.3 The working window of the program quadratic equation.

A quadratic equation is an equation of the form a*x^2 +b*x+c=0, where a,b,c are some arbitrary real (real) numbers, and x is a variable. And the number a=0.

The numbers a,b,c are called coefficients. The number a - is called the leading coefficient, the number b is the coefficient at x, and the number c is called the free member.

Solving quadratic equations

To solve a quadratic equation means to find all its roots, or to establish the fact that the quadratic equation has no roots. The root of the quadratic equation a * x ^ 2 + b * x + c \u003d 0 is any value of the variable x, such that square trinomial a*x^2 +b*x+c vanishes. Sometimes such a value of x is called the root of a square trinomial.

There are several ways to solve quadratic equations. Consider one of them - the most versatile. It can be used to solve any quadratic equation.

Formulas for solving quadratic equations

The formula for the roots of the quadratic equation is a*x^2 +b*x+c=0.

x=(-b±√D)/(2*a), where D =b^2-4*a*c.

This formula is obtained by solving the equation a * x ^ 2 + b * x + c \u003d 0 in general form, by highlighting the square of the binomial.

In the formula of the roots of a quadratic equation, the expression D (b^2-4*a*c) is called the discriminant of the quadratic equation a*x^2 +b*x+c=0. This name came from Latin, in translation "distinguisher". Depending on the value of the discriminant, the quadratic equation will have two or one root, or no roots at all.

If the discriminant is greater than zero, then the quadratic equation has two roots. (x=(-b±√D)/(2*a))

If the discriminant is zero, then the quadratic equation has one root. (x=(-b/(2*a))

If the discriminant is negative, then the quadratic equation has no roots.

General algorithm for solving a quadratic equation

Based on the foregoing, we formulate a general algorithm for solving the quadratic equation a*x^2 +b*x+c=0 using the formula:

1. Find the value of the discriminant using the formula D =b^2-4*a*c.

2. Depending on the value of the discriminant, calculate the roots using the formulas:

D<0, корней нет.

D=0, x=(-b/(2*a)

D>0, x=(-b+√D)/(2*a), x=(-b-√D)/(2*a)

This algorithm is universal and suitable for solving any quadratic equations. Complete and incomplete, cited and not cited.