How to find the discriminant of an incomplete quadratic equation. Quadratic equation

The transformation of a complete quadratic equation into an incomplete one looks like this (for the case \(b=0\)):

For cases when \(c=0\) or when both coefficients are equal to zero, everything is similar.

Please note that there is no question of \(a\) being equal to zero; it cannot be equal to zero, since in this case it will turn into :

Solving incomplete quadratic equations.

First of all, you need to understand that an incomplete quadratic equation is still a , and therefore can be solved in the same way as an ordinary quadratic equation (via ). To do this, we simply add the missing component of the equation with a zero coefficient.

Example : Find the roots of the equation \(3x^2-27=0\)
Solution :

		We have an incomplete quadratic equation with coefficient \(b=0\). That is, we can write the equation in the following form:
\(3x^2+0\cdot x-27=0\)		In fact, this is the same equation as in the beginning, but now it can be solved as an ordinary quadratic one. First we write out the coefficients.
\(a=3;\) \(b=0;\) \(c=-27;\)		Let's calculate the discriminant using the formula \(D=b^2-4ac\)
\(D=0^2-4\cdot3\cdot(-27)=\) \(=0+324=324\)		Let's find the roots of the equation using the formulas \(x_(1)=\)\(\frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b-\sqrt(D) )(2a)\)
\(x_(1)=\) \(\frac(-0+\sqrt(324))(2\cdot3)\)\(=\)\(\frac(18)(6)\) \(=3\) \(x_(2)=\) \(\frac(-0-\sqrt(324))(2\cdot3)\)\(=\)\(\frac(-18)(6)\) \(=-3\)		Write down the answer

Answer : \(x_(1)=3\); \(x_(2)=-3\)

Example : Find the roots of the equation \(-x^2+x=0\)
Solution :

		Again an incomplete quadratic equation, but now the coefficient \(c\) is equal to zero. We write the equation as complete.

From this article you will learn:

What is it in appearance equations determine whether this equation will be incomplete quadratic equation? But as solve incomplete quadratic equations?

How to recognize an incomplete quadratic equation by sight

Left part of the equation There is quadratic trinomial, A right - number. Such equations are called full quadratic equations.

U full quadratic equation All odds, And not equal. To solve them, there are special formulas, which we will get acquainted with later.

Most simple for solution are incomplete quadratic equations. These are quadratic equations in which some coefficients are zero.

Coefficient by definition cannot be zero, since otherwise the equation will not be quadratic. We talked about this. This means that it turns out that they can go to zero only odds or.

Depending on this there is three types of incomplete quadratic equations.

1) , Where ;
2) , Where ;
3) , Where .

So, if we see a quadratic equation, on the left side of which instead of three members present two dicks or one member, then the equation will be incomplete quadratic equation.

Definition of an incomplete quadratic equation

Incomplete quadratic equation This is called a quadratic equation , in which at least one of the coefficients or equal to zero.

This definition has a lot important phrase " at least one from the coefficients... equal to zero". It means that one or more coefficients can be equal zero.

Based on this, it is possible three options: or one coefficient is zero, or another coefficient is zero, or both coefficients are simultaneously equal to zero. This is how we get three types of incomplete quadratic equations.

Incomplete quadratic equations are the following equations:
1)
2)
3)

Solving the equation

Let's outline solution plan this equation. Left part of the equation can be easily factorize, since on the left side of the equation the terms have common multiplier, it can be taken out of the bracket. Then on the left you get the product of two factors, and on the right - zero.

And then the rule “the product is equal to zero if and only if at least one of the factors is equal to zero, and the other makes sense” will work. Everything is very simple!

So, solution plan.
1) We factor the left side into factors.
2) We use the rule “the product is equal to zero...”

Equations similar type I call "a gift of fate". These are equations for which the right side is zero, A left part can be expanded by multipliers.

Solving the equation according to plan.

1) Let's decompose left side equations by multipliers, for this we take out the common factor, we get the following equation .

2) In Eq. we see that left costs work, A zero on the right. Real a gift of fate! Here we, of course, will use the rule “the product is equal to zero if and only if at least one of the factors is equal to zero, and the other makes sense.” When translating this rule into the language of mathematics, we get two equations or .

We see that the equation fell apart by two simpler equations, the first of which has already been solved ().

Let's solve the second one the equation . Let's move the unknown terms to the left and the known ones to the right. The unknown member is already on the left, we will leave him there. And we move the known term to the right with the opposite sign. We get the equation.

We found it, but we need to find it. To get rid of the factor, you need to divide both sides of the equation by.

An incomplete quadratic equation differs from classical (complete) equations in that its factors or free term are equal to zero. The graphs of such functions are parabolas. Depending on their general appearance, they are divided into 3 groups. The principles of solution for all types of equations are the same.

There is nothing complicated in determining the type of an incomplete polynomial. It is best to consider the main differences using visual examples:

1. If b = 0, then the equation is ax 2 + c = 0.
2. If c = 0, then the expression ax 2 + bx = 0 should be solved.
3. If b = 0 and c = 0, then the polynomial turns into an equality like ax 2 = 0.

The latter case is more of a theoretical possibility and never occurs in knowledge testing tasks, since the only correct value of the variable x in the expression is zero. In the future, methods and examples of solving incomplete quadratic equations of 1) and 2) types will be considered.

General algorithm for searching variables and examples with solutions

Regardless of the type of equation, the solution algorithm is reduced to the following steps:

1. Reduce the expression to a form convenient for finding roots.
2. Perform calculations.
3. Write down the answer.

The easiest way to solve incomplete equations is to factor the left side and leave a zero on the right. Thus, the formula for an incomplete quadratic equation for finding roots is reduced to calculating the value of x for each of the factors.

You can only learn how to solve it in practice, so let’s consider specific example finding the roots of an incomplete equation:

As you can see, in this case b = 0. Let’s factorize the left side and get the expression:

4(x – 0.5) ⋅ (x + 0.5) = 0.

Obviously, the product is equal to zero when at least one of the factors is equal to zero. The values ​​of the variable x1 = 0.5 and (or) x2 = -0.5 meet similar requirements.

In order to easily and quickly cope with the problem of factoring a quadratic trinomial, you should remember the following formula:

If there is no free term in the expression, the problem is greatly simplified. It will be enough just to find and bracket the common denominator. For clarity, consider an example of how to solve incomplete quadratic equations of the form ax2 + bx = 0.

Let's take the variable x out of brackets and get the following expression:

x ⋅ (x + 3) = 0.

Guided by logic, we come to the conclusion that x1 = 0, and x2 = -3.

Traditional solution method and incomplete quadratic equations

What happens if you apply the discriminant formula and try to find the roots of a polynomial with coefficients equal to zero? Let's take an example from a collection of standard tasks for the Unified State Exam in Mathematics 2017, solve it using standard formulas and the factorization method.

7x 2 – 3x = 0.

Let's calculate the discriminant value: D = (-3)2 – 4 ⋅ (-7) ⋅ 0 = 9. It turns out that the polynomial has two roots:

Now, let's solve the equation by factoring and compare the results.

X ⋅ (7x + 3) = 0,

2) 7x + 3 = 0,
7x = -3,
x = -.

As you can see, both methods give the same result, but solving the equation using the second method was much easier and faster.

Vieta's theorem

But what to do with Vieta’s favorite theorem? Is it possible to use this method with an incomplete trinomial? Let's try to understand the aspects of casting complete equations To classic look ax2 + bx + c = 0.

In fact, it is possible to apply Vieta's theorem in this case. It is only necessary to bring the expression to its general form, replacing the missing terms with zero.

For example, with b = 0 and a = 1, in order to eliminate the possibility of confusion, the task should be written in the form: ax2 + 0 + c = 0. Then the ratio of the sum and product of the roots and factors of the polynomial can be expressed as follows:

Theoretical calculations help to get acquainted with the essence of the issue, and always require the development of skills when solving specific problems. Let's turn again to the reference book of standard tasks for the Unified State Exam and find a suitable example:

Let us write the expression in a form convenient for applying Vieta’s theorem:

x 2 + 0 – 16 = 0.

The next step is to create a system of conditions:

Obviously, the roots of the quadratic polynomial will be x 1 = 4 and x 2 = -4.

Now, let's practice bringing the equation to its general form. Let's take the following example: 1/4× x 2 – 1 = 0

In order to apply Vieta's theorem to an expression, it is necessary to get rid of the fraction. Let’s multiply the left and right sides by 4, and look at the result: x2– 4 = 0. The resulting equality is ready to be solved by Vieta’s theorem, but it is much easier and faster to get the answer by simply moving c = 4 to the right side of the equation: x2 = 4.

To summarize, it should be said that the best way solving incomplete equations is factorization, is the simplest and quick method. If difficulties arise in the process of searching for roots, you can contact traditional method finding roots through a discriminant.

It is known that it is a particular version of the equality ax 2 + bx + c = o, where a, b and c are real coefficients for unknown x, and where a ≠ o, and b and c will be zeros - simultaneously or separately. For example, c = o, b ≠ o or vice versa. We almost remembered the definition of a quadratic equation.

The second degree trinomial is zero. Its first coefficient a ≠ o, b and c can take any values. The value of the variable x will then be when substitution turns it into a correct numerical equality. Let's focus on real roots, although the equations can also be solutions. It is customary to call an equation complete in which none of the coefficients is equal to o, a ≠ o, b ≠ o, c ≠ o.
Let's solve an example. 2x 2 -9x-5 = oh, we find
D = 81+40 = 121,
D is positive, which means there are roots, x 1 = (9+√121):4 = 5, and the second x 2 = (9-√121):4 = -o.5. Checking will help make sure they are correct.

Here step by step solution quadratic equation

Using the discriminant, you can solve any equation on the left side of which there is a known quadratic trinomial for a ≠ o. In our example. 2x 2 -9x-5 = 0 (ax 2 +in+s = o)

Let's consider what incomplete equations of the second degree are

1. ax 2 +in = o. The free term, the coefficient c at x 0, is equal to zero here, in ≠ o.
   How to solve an incomplete quadratic equation of this type? Let's take x out of brackets. Let's remember when the product of two factors is equal to zero.
   x(ax+b) = o, this can be when x = o or when ax+b = o.
   Having solved the 2nd we have x = -в/а.
   As a result, we have roots x 1 = 0, according to calculations x 2 = -b/a.
2. Now the coefficient of x is equal to o, and c is not equal (≠) o.
   x 2 +c = o. Let's move c to the right side of the equality, we get x 2 = -с. This equation only has real roots when -c positive number(with ‹ o),
   x 1 is then equal to √(-c), respectively, x 2 is -√(-c). Otherwise, the equation has no roots at all.
3. The last option: b = c = o, that is, ax 2 = o. Naturally, such a simple equation has one root, x = o.

Special cases

We looked at how to solve an incomplete quadratic equation, and now let’s take any types.

• In a complete quadratic equation, the second coefficient of x is an even number.
  Let k = o.5b. We have formulas for calculating the discriminant and roots.
  D/4 = k 2 - ac, the roots are calculated as x 1,2 = (-k±√(D/4))/a for D › o.
  x = -k/a at D = o.
  There are no roots for D ‹ o.
• There are given quadratic equations, when the coefficient of x squared is equal to 1, they are usually written x 2 + рх + q = o. All the above formulas apply to them, but the calculations are somewhat simpler.
  Example, x 2 -4x-9 = 0. Calculate D: 2 2 +9, D = 13.
  x 1 = 2+√13, x 2 = 2-√13.
• In addition, it is easy to apply to those given. It says that the sum of the roots of the equation is equal to -p, the second coefficient with a minus (meaning the opposite sign), and the product of these same roots will be equal to q, the free term. See how easy it would be to determine the roots of this equation verbally. For unreduced coefficients (for all coefficients not equal to zero), this theorem is applicable as follows: the sum x 1 + x 2 is equal to -b/a, the product x 1 · x 2 is equal to c/a.

The sum of the free term c and the first coefficient a is equal to the coefficient b. In this situation, the equation has at least one root (easy to prove), the first one is necessarily equal to -1, and the second -c/a, if it exists. You can check how to solve an incomplete quadratic equation yourself. As easy as pie. The coefficients may be in certain relationships with each other

• x 2 +x = o, 7x 2 -7 = o.
• The sum of all coefficients is equal to o.
  The roots of such an equation are 1 and c/a. Example, 2x 2 -15x+13 = o.
  x 1 = 1, x 2 = 13/2.

There are a number of other ways to solve various second-degree equations. Here, for example, is a method for extracting from a given polynomial full square. There are several graphical methods. When you often deal with such examples, you will learn to “click” them like seeds, because all the methods come to mind automatically.

First level

Quadratic equations. The Comprehensive Guide (2019)

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

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

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

Example 1.

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

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

Now we can say with confidence that this equation is quadratic!

Example 2.

Multiply the left and right sides by:

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

Example 3.

Let's multiply everything by:

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

Example 4.

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

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

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

Examples:

Answers:

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

Mathematicians conventionally divide all quadratic equations into the following types:

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

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

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

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

Solving incomplete quadratic equations

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

There are types of incomplete quadratic equations:

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

1. i. Since we know how to take the square root, let's express from this equation

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

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

Let's try to solve some examples.

Example 5:

Solve the equation

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

Answer:

Never forget about roots with a negative sign!!!

Example 6:

Solve the equation

Answer:

Example 7:

Solve the equation

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

no roots!

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

Answer:

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

Solve the equation

Let's take the common factor out of brackets:

Thus,

This equation has two roots.

Answer:

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

We will dispense with examples here.

Solving complete quadratic equations

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

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

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

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

1. Solving quadratic equations using a discriminant.

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

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

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

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

Example 9:

Solve the equation

Step 1 we skip.

Step 2.

We find the discriminant:

This means the equation has two roots.

Step 3.

Answer:

Example 10:

Solve the equation

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

Step 2.

We find the discriminant:

This means that the equation has one root.

Answer:

Example 11:

Solve the equation

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

Step 2.

We find the discriminant:

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

Now we know how to correctly write down such answers.

Answer: no roots

2. Solving quadratic equations using Vieta's theorem.

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

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

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

Example 12:

Solve the equation

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

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

And the product is equal to:

Let's compose and solve the system:

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

and are the solution to the system:

Answer: ; .

Example 13:

Solve the equation

Answer:

Example 14:

Solve the equation

The equation is given, which means:

Answer:

QUADRATIC EQUATIONS. AVERAGE LEVEL

What is a quadratic equation?

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

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

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

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

Solutions to various types of quadratic equations

Methods for solving incomplete quadratic equations:

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

We can distinguish the following types of equations:

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

II. , in this equation the coefficient is equal.

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

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

Obviously, this equation always has only one root:

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

if, then the equation has no solutions;

if we have two roots

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

Examples:

Solutions:

Answer:

Never forget about roots with a negative sign!

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

no roots.

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

Answer:

So, this equation has two roots: and.

Answer:

Let's take the common factor out of brackets:

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

So, this quadratic equation has two roots: and.

Example:

Solve the equation.

Solution:

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

Answer:

Methods for solving complete quadratic equations:

1. Discriminant

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

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

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

  Such roots are called double roots.

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

Why is it possible different quantities roots? Let 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.

Examples:

Solutions:

Answer:

Answer: .

Answer:

This means there are no solutions.

Answer: .

2. Vieta's theorem

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

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

Let's look at a few examples:

Example #1:

Solve the equation.

Solution:

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

The sum of the roots of the equation is:

And the product is equal to:

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

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

and are the solution to the system:

Thus, and are the roots of our equation.

Answer: ; .

Example #2:

Solution:

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

and: they give in total.

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

Answer:

Example #3:

Solution:

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

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

and: their difference is equal - does not fit;

and: - not suitable;

and: - not suitable;

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

Answer:

Example #4:

Solve the equation.

Solution:

The equation is given, which means:

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

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

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

Answer:

Example #5:

Solve the equation.

Solution:

The equation is given, which means:

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

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

Obviously, the roots are the numbers and.

Answer:

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

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

Solutions to tasks for independent work:

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

According to Vieta's theorem:

As usual, we start the selection with the piece:

Not suitable because the amount;

: the amount is just what you need.

Answer: ; .

Task 2.

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

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

Answer: ; .

Task 3.

Hmm... Where is that?

You need to move all the terms into one part:

The sum of the roots is equal to the product.

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

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

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

Answer: ; .

Task 4.

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

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

Answer: ; .

Task 5.

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

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

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

Answer: ; .

Let me summarize:

1. Vieta's theorem is used only in the quadratic equations given.
2. Using Vieta's theorem, you can find the roots by selection, orally.
3. If the equation is not given or no 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 1:

Solve the equation: .

Solution:

Answer:

Example 2:

Solve the equation: .

Solution:

Answer:

IN general view the transformation will look like this:

This implies: .

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

QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN THINGS

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

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

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

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

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

1. Algorithm for solving incomplete quadratic equations

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

1) Let's express the unknown: ,

2) Check the sign of the expression:

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

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

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

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

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

This equation always has only one root: .

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

2.1. Solution using discriminant

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

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

3) Find the roots of the equation:

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

2.2. Solution using Vieta's theorem

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

2.3. Solution by the method of selecting a complete square