Methods for solving quadratic equations. Computer Science Lesson "Solving a Quadratic Equation"
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:
- Have no roots;
- They have exactly one root;
- They have two different roots.
This is what important difference quadratic equations from linear ones, 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:
- If D< 0, корней нет;
- If D = 0, there is exactly one root;
- 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:
- x 2 - 8x + 12 = 0;
- 5x2 + 3x + 7 = 0;
- 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:
Basic formula of roots 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.
- x 2 - 2x - 3 = 0;
- 15 - 2x - x2 = 0;
- 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:
- x2 + 9x = 0;
- 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:
Because arithmetic Square root exists only from a non-negative number, the last equality makes sense only for (−c /a ) ≥ 0. Conclusion:
- 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;
- If (−c / a )< 0, корней нет.
As you can see, the discriminant was not required - in incomplete quadratic equations there is no complex calculations. 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 is a 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 bracketThe 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:
- x2 − 7x = 0;
- 5x2 + 30 = 0;
- 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.
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. You can use a simple program for this. 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 working window quadratic equation program.
The problem is well known from mathematics. The initial data here are the coefficients a, b, c. Decision in general case are two roots x 1 and x 2 , which are calculated by the formulas:
All values used in this program are of real type.
alg roots of a quadratic equation
thing a, b, c, x1, x2, d
early input a, b, c
x1:=(-b+Öd)/(2a)
x2:=(-b–Öd)/(2a)
output x1, x2
The weakness of such an algorithm is visible to the naked eye. It does not have the most important property required for high-quality algorithms: universality with respect to the initial data. Whatever the values of the initial data, the algorithm must lead to a certain result and reach the end. The result may be a numerical answer, but it may also be a message that with such data the problem has no solution. Stops in the middle of the algorithm due to the impossibility of performing some operation are not allowed. The same property in the literature on programming is called the effectiveness of the algorithm (in any case, some result must be obtained).
In order to build a universal algorithm, it is first necessary to carefully analyze the mathematical content of the problem.
The solution of the equation depends on the values of the coefficients a, b, c. Here is an analysis of this problem (we restrict ourselves only to finding real roots):
if a=0, b=0, c=0, then any x is a solution to the equation;
if a=0, b=0, c¹0, then the equation has no solutions;
if a=0, b¹0, then this is a linear equation that has one solution: x=–c/b;
if a¹0 and d=b 2 -4ac³0, then the equation has two real roots (the formulas are given above);
if a¹0 and d<0, то уравнение не имеет вещественных корней.
Block diagram of the algorithm:
The same algorithm in algorithmic language:
alg roots of a quadratic equation
thing a, b, c, d, x1, x2
early input a, b, c
if a=0
then if b=0
then if c=0
then output "any x is a solution"
otherwise output "no solutions"
otherwise x:= -c/b
otherwise d:=b2–4ac
if and d<0
then output "no real roots"
otherwise e x1:=(-b+Öd)/(2a); x2:=(-b–Öd)/(2a)
output “x1=”,x1, “x2=”,x2
This algorithm reuses branch structure command. The general view of the branch command in flowcharts and in the algorithmic language is as follows:
First, the “condition” is checked (the relation, the logical expression is calculated). If the condition is true, then "series 1" is executed - the sequence of commands indicated by the arrow with the inscription "yes" (positive branch). Otherwise, "series 2" (negative branch) is executed. In EL, the condition is written after the service word "if", the positive branch - after the word "then", the negative branch - after the word "otherwise". The letters "kv" indicate the end of the branch.
If the branches of one branch contain other branches, then such an algorithm has the structure nested branches. It is this structure that the algorithm "roots of a quadratic equation" has. In it, for brevity, instead of the words "yes" and "no", respectively, "+" and "-" are used.
Consider the following problem: given a positive integer n. It is required to calculate n! (n-factorial). Recall the definition of factorial.
Below is a block diagram of the algorithm. It uses three integer type variables: n is an argument; i is an intermediate variable; F is the result. A trace table was built to check the correctness of the algorithm. In such a table, for specific values of the initial data, the changes in the variables included in the algorithm are traced by steps. This table is compiled for the case n=3.
The trace proves the correctness of the algorithm. Now let's write this algorithm in algorithmic language.
alg Factorial
whole n, i, F
early input n
F:=1; i:=1
Bye i£n, repeat
nc F:=F´i
This algorithm has a cyclic structure. The algorithm uses the "loop-while" or "loop with precondition" structural command. The general view of the “loop-bye” command in flowcharts and in EL is as follows:
The execution of a series of commands (loop body) is repeated while the loop condition is true. When the condition becomes false, the loop terminates. The service words "nts" and "kts" denote the beginning of the cycle and the end of the cycle, respectively.
A loop with a precondition is the main, but not the only form of organization of cyclic algorithms. Another option is loop with postcondition. Let's return to the algorithm for solving a quadratic equation. It can be approached from this position: if a=0, then this is no longer a quadratic equation and it can be ignored. In this case, we will assume that the user made a mistake when entering data and should be prompted to repeat the entry. In other words, the algorithm will provide for the control of the reliability of the initial data, providing the user with the opportunity to correct the error. The presence of such control is another sign of good program quality.
In general, the structural command "loop with postcondition" or "loop-before" is represented as follows:
This is where the loop termination condition is used. When it becomes true, the loop terminates.
Let us compose an algorithm for solving the following problem: given two natural numbers M and N. It is required to calculate their greatest common divisor - gcd(M,N).
This problem is solved using a method known as Euclid's algorithm. His idea is based on the property that if M>N, then gcd(M 1) if the numbers are equal, then take their total value as an answer; otherwise, continue the execution of the algorithm; 2) determine the larger of the numbers; 3) replace the larger number with the difference between the larger and smaller values; 4) return to the implementation of paragraph 1. The block diagram and algorithm in AL will be as follows: The algorithm has a loop structure with nested branching. Do your own tracing of this algorithm for the case M=18, N=12. The result is gcd=6, which is obviously true. 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 earthworks of a 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. Using the 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. Despite the high level of development of algebra in Babylonia, the concept of a negative number and general methods for solving quadratic equations are absent in cuneiform texts. 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. slide 12 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. slide 13 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 Slide 14 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 slide 15 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. Important notes! 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 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 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: Mathematicians conditionally divide all quadratic equations into the following types: 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. First, let's focus on solving incomplete quadratic equations - they are much simpler! Incomplete quadratic equations are of types: 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. 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. 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. 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. 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 should be paid to the step. The discriminant () tells us the number of roots of the equation. 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 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 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: 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. 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 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. Decision: We factorize the left side of the equation and find the roots: Answer: 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. Such roots are called double 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: . 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 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: ; . 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: 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 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: 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: 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: 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. 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Example 8:Solving complete quadratic equations
1. Solving quadratic equations using the discriminant.
2. Solution of quadratic equations using the Vieta theorem.
QUADRATIC EQUATIONS. MIDDLE LEVEL
What is a quadratic equation?
Solutions to various types of quadratic equations
Methods for solving incomplete quadratic equations:
Methods for solving complete quadratic equations:
1. Discriminant
2. Vieta's theorem
Let me summarize:
3. Full square selection method
QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN