Solving quadratic equations, root formula, examples. Quadratic equations. Solving Quadratic Equations How to Convert a Quadratic Equation to a Product
This topic may seem difficult at first due to many not so simple formulas. Not only do the quadratic equations themselves have long notations, but the roots are also found through the discriminant. In total, three new formulas are obtained. Not very easy to remember. This is possible only after solving such equations frequently. Then all the formulas will be remembered by themselves.
General view of a quadratic equation
Here we propose their explicit recording, when the largest degree is written first, and then in descending order. There are often situations when the terms are inconsistent. Then it is better to rewrite the equation in descending order of the degree of the variable.
Let us introduce some notation. They are presented in the table below.
If we accept these notations, all quadratic equations are reduced to the following notation.
Moreover, the coefficient a ≠ 0. Let this formula be designated number one.
When an equation is given, it is not clear how many roots there will be in the answer. Because one of three options is always possible:
- the solution will have two roots;
- the answer will be one number;
- the equation will have no roots at all.
And until the decision is finalized, it is difficult to understand which option will appear in a particular case.
Types of recordings of quadratic equations
There may be different entries in tasks. They won't always look like general formula quadratic equation. Sometimes it will be missing some terms. What was written above is the complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.
Moreover, only terms with coefficients “b” and “c” can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of equations will be as follows:
So, there are only two types; in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second - three.
Discriminant and dependence of the number of roots on its value
You need to know this number in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have number four.
After substituting the coefficient values into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. If the number is negative, there will be no roots of the quadratic equation. If it is equal to zero, there will be only one answer.
How to solve a complete quadratic equation?
In fact, consideration of this issue has already begun. Because first you need to find a discriminant. After it is determined that there are roots of the quadratic equation, and their number is known, you need to use formulas for the variables. If there are two roots, then you need to apply the following formula.
Since it contains a “±” sign, there will be two values. The expression under the square root sign is the discriminant. Therefore, the formula can be rewritten differently.
Formula number five. From the same record it is clear that if the discriminant is equal to zero, then both roots will take the same values.
If solving quadratic equations has not yet been worked out, then it is better to write down the values of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.
How to solve an incomplete quadratic equation?
Everything is much simpler here. There is not even a need for additional formulas. And those that have already been written down for the discriminant and the unknown will not be needed.
First, let's look at incomplete equation number two. In this equality, it is necessary to take the unknown quantity out of brackets and solve the linear equation, which will remain in brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a multiplier consisting of the variable itself. The second one will be obtained by solving a linear equation.
Incomplete equation number three is solved by moving the number from the left side of the equality to the right. Then you need to divide by the coefficient facing the unknown. All that remains is to extract the square root and remember to write it down twice with opposite signs.
Below are some steps that will help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings can cause poor grades when studying the extensive topic “Quadratic Equations (8th Grade).” Subsequently, these actions will not need to be performed constantly. Because a stable skill will appear.
- First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without a degree, and last - just a number.
- If a minus appears before the coefficient “a”, it can complicate the work for a beginner studying quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by “-1”. This means that all terms will change sign to the opposite.
- It is recommended to get rid of fractions in the same way. Simply multiply the equation by the appropriate factor so that the denominators cancel out.
Examples
It is required to solve the following quadratic equations:
x 2 − 7x = 0;
15 − 2x − x 2 = 0;
x 2 + 8 + 3x = 0;
12x + x 2 + 36 = 0;
(x+1) 2 + x + 1 = (x+1)(x+2).
The first equation: x 2 − 7x = 0. It is incomplete, therefore it is solved as described for formula number two.
After taking it out of brackets, it turns out: x (x - 7) = 0.
The first root takes the value: x 1 = 0. The second will be found from the linear equation: x - 7 = 0. It is easy to see that x 2 = 7.
Second equation: 5x 2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.
After moving 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be the numbers: x 1 = √6, x 2 = - √6.
The third equation: 15 − 2x − x 2 = 0. Here and further, solving quadratic equations will begin by rewriting them in standard form: − x 2 − 2x + 15 = 0. Now it’s time to use the second useful advice and multiply everything by minus one. It turns out x 2 + 2x - 15 = 0. Using the fourth formula, you need to calculate the discriminant: D = 2 2 - 4 * (- 15) = 4 + 60 = 64. It is a positive number. From what is said above, it turns out that the equation has two roots. They need to be calculated using the fifth formula. It turns out that x = (-2 ± √64) / 2 = (-2 ± 8) / 2. Then x 1 = 3, x 2 = - 5.
The fourth equation x 2 + 8 + 3x = 0 is transformed into this: x 2 + 3x + 8 = 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: “There are no roots.”
The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x = -12/ (2 * 1) = -6.
The sixth equation (x+1) 2 + x + 1 = (x+1)(x+2) requires transformations, which consist in the fact that you need to bring similar terms, first opening the brackets. In place of the first there will be the following expression: x 2 + 2x + 1. After the equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x = 0. It has become incomplete . Something similar to this has already been discussed a little higher. The roots of this will be the numbers 0 and 1.
Some problems in mathematics require the ability to calculate the value of the square root. Such problems include solving second-order equations. In this article we present an effective method for calculating square roots and use it when working with formulas for the roots of a quadratic equation.
What is a square root?
In mathematics, this concept corresponds to the symbol √. Historical data says that it was first used around the first half of the 16th century in Germany (the first German work on algebra by Christoph Rudolf). Scientists believe that the symbol is a transformed Latin letter r (radix means "root" in Latin).
The root of any number is equal to the value whose square corresponds to the radical expression. In the language of mathematics, this definition will look like this: √x = y, if y 2 = x.
The root of a positive number (x > 0) is also a positive number (y > 0), but if we take the root of negative number(x< 0), то его результатом уже будет complex number, including the imaginary unit i.
Here are two simple examples:
√9 = 3, since 3 2 = 9; √(-9) = 3i, since i 2 = -1.
Heron's iterative formula for finding the values of square roots
The above examples are very simple, and calculating the roots in them is not difficult. Difficulties begin to appear when finding root values for any value that cannot be represented as a square natural number, for example √10, √11, √12, √13, not to mention the fact that in practice it is necessary to find roots for non-integer numbers: for example √(12,15), √(8,5) and so on.
In all of the above cases, a special method for calculating the square root should be used. Currently, several such methods are known: for example, Taylor series expansion, column division and some others. Of all the known methods, perhaps the simplest and most effective is the use of Heron's iterative formula, which is also known as the Babylonian method of determining square roots (there is evidence that the ancient Babylonians used it in their practical calculations).
Let it be necessary to determine the value of √x. The formula for finding the square root is as follows:
a n+1 = 1/2(a n +x/a n), where lim n->∞ (a n) => x.
Let's decipher this mathematical notation. To calculate √x, you should take a certain number a 0 (it can be arbitrary, but to quickly get the result, you should choose it so that (a 0) 2 is as close as possible to x. Then substitute it into the indicated formula for calculating the square root and get a new number a 1, which will already be closer to the desired value. After this, you need to substitute a 1 into the expression and get a 2. This procedure should be repeated until the required accuracy is obtained.
An example of using Heron's iterative formula
The algorithm described above for obtaining the square root of a given number may sound quite complicated and confusing to many, but in reality everything turns out to be much simpler, since this formula converges very quickly (especially if a successful number a 0 is chosen).
Let's give a simple example: you need to calculate √11. Let's choose a 0 = 3, since 3 2 = 9, which is closer to 11 than 4 2 = 16. Substituting into the formula, we get:
a 1 = 1/2(3 + 11/3) = 3.333333;
a 2 = 1/2(3.33333 + 11/3.33333) = 3.316668;
a 3 = 1/2(3.316668 + 11/3.316668) = 3.31662.
There is no point in continuing the calculations, since we found that a 2 and a 3 begin to differ only in the 5th decimal place. Thus, it was enough to apply the formula only 2 times to calculate √11 with an accuracy of 0.0001.
Nowadays, calculators and computers are widely used to calculate roots, however, it is useful to remember the marked formula in order to be able to manually calculate their exact value.
Second order equations
Understanding what a square root is and the ability to calculate it is used in solving quadratic equations. These equations are called equalities with one unknown, the general form of which is shown in the figure below.
Here c, b and a represent some numbers, and a must not be equal to zero, and the values of c and b can be completely arbitrary, including equal to zero.
Any values of x that satisfy the equality indicated in the figure are called its roots (this concept should not be confused with the square root √). Since the equation under consideration is of the 2nd order (x 2), then there cannot be more than two roots for it. Let's look further in the article at how to find these roots.
Finding the roots of a quadratic equation (formula)
This method of solving the type of equalities under consideration is also called the universal method, or the discriminant method. It can be used for any quadratic equations. The formula for the discriminant and roots of the quadratic equation is as follows:
It shows that the roots depend on the value of each of the three coefficients of the equation. Moreover, the calculation of x 1 differs from the calculation of x 2 only by the sign in front of the square root. The radical expression, which is equal to b 2 - 4ac, is nothing more than the discriminant of the equality in question. The discriminant in the formula for the roots of a quadratic equation plays an important role because it determines the number and type of solutions. So, if it is equal to zero, then there will be only one solution, if it is positive, then the equation has two real roots, and finally, a negative discriminant leads to two complex roots x 1 and x 2.
Vieta's theorem or some properties of the roots of second-order equations
At the end of the 16th century, one of the founders of modern algebra, a Frenchman, studying second-order equations, was able to obtain the properties of its roots. Mathematically they can be written like this:
x 1 + x 2 = -b / a and x 1 * x 2 = c / a.
Both equalities can be easily obtained by anyone; to do this, you just need to perform the appropriate mathematical operations with the roots obtained through the formula with the discriminant.
The combination of these two expressions can rightly be called the second formula for the roots of a quadratic equation, which makes it possible to guess its solutions without using a discriminant. Here it should be noted that although both expressions are always valid, it is convenient to use them to solve an equation only if it can be factorized.
The task of consolidating the acquired knowledge
Let's decide math problem, in which we will demonstrate all the techniques discussed in the article. The conditions of the problem are as follows: you need to find two numbers for which the product is -13 and the sum is 4.
This condition immediately reminds us of Vieta’s theorem; using the formulas for the sum of square roots and their product, we write:
x 1 + x 2 = -b / a = 4;
x 1 * x 2 = c / a = -13.
If we assume that a = 1, then b = -4 and c = -13. These coefficients allow us to create a second-order equation:
x 2 - 4x - 13 = 0.
Let's use the formula with the discriminant and get the following roots:
x 1.2 = (4 ± √D)/2, D = 16 - 4 * 1 * (-13) = 68.
That is, the problem was reduced to finding the number √68. Note that 68 = 4 * 17, then, using the square root property, we get: √68 = 2√17.
Now let’s use the considered square root formula: a 0 = 4, then:
a 1 = 1/2(4 + 17/4) = 4.125;
a 2 = 1/2(4.125 + 17/4.125) = 4.1231.
There is no need to calculate a 3 since the values found differ by only 0.02. Thus, √68 = 8.246. Substituting it into the formula for x 1,2, we get:
x 1 = (4 + 8.246)/2 = 6.123 and x 2 = (4 - 8.246)/2 = -2.123.
As we can see, the sum of the numbers found is really equal to 4, but if we find their product, then it will be equal to -12.999, which satisfies the conditions of the problem with an accuracy of 0.001.
Just. According to formulas and clear, simple rules. At the first stage
it is necessary to bring the given equation to a standard form, i.e. to the form:
If the equation is already given to you in this form, you do not need to do the first stage. The most important thing is to do it right
determine all the coefficients, A, b And c.
Formula for finding the roots of a quadratic equation.
The expression under the root sign is called discriminant . As you can see, to find X, we
we use only a, b and c. Those. coefficients from quadratic equation. Just carefully put it in
values a, b and c We calculate into this formula. We substitute with their signs!
For example, in the equation:
A =1; b = 3; c = -4.
We substitute the values and write:
The example is almost solved:
This is the answer.
The most common mistakes are confusion with sign values a, b And With. Or rather, with substitution
negative values into the formula for calculating the roots. A detailed recording of the formula comes to the rescue here
with specific numbers. If you have problems with calculations, do it!
Suppose we need to solve the following example:
Here a = -6; b = -5; c = -1
We describe everything in detail, carefully, without missing anything with all the signs and brackets:
Quadratic equations often look slightly different. For example, like this:
Now take note of practical techniques that dramatically reduce the number of errors.
First appointment. Don't be lazy before solving a quadratic equation bring it to standard form.
What does this mean?
Let's say that after all the transformations you get the following equation:
Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c.
Construct the example correctly. First, X squared, then without square, then the free term. Like this:
Get rid of the minus. How? We need to multiply the entire equation by -1. We get:
But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example.
Decide for yourself. You should now have roots 2 and -1.
Reception second. Check the roots! By Vieta's theorem.
To solve the given quadratic equations, i.e. if the coefficient
x 2 +bx+c=0,
Thenx 1 x 2 =c
x 1 +x 2 =−b
For a complete quadratic equation in which a≠1:
x 2 +bx+c=0,
divide the whole equation by A:
→ 
→
Where x 1 And x 2 - roots of the equation.
Reception third. If your equation has fractional coefficients, get rid of the fractions! Multiply
equation with a common denominator.
Conclusion. Practical tips:
1. Before solving, we bring the quadratic equation to standard form and build it Right.
2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying everything
equations by -1.
3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding
factor.
4. If x squared is pure, its coefficient is equal to one, the solution can be easily checked by
Solving equations in mathematics occupies a special place. This process is preceded by many hours of studying theory, during which the student learns how to solve equations, determine their type, and brings the skill to complete automation. However, searching for roots does not always make sense, since they may simply not exist. There are special techniques for finding roots. In this article we will analyze the main functions, their domains of definition, as well as cases when their roots are missing.
Which equation has no roots?
An equation has no roots if there are no real arguments x for which the equation is identically true. For a non-specialist, this formulation, like most mathematical theorems and formulas, looks very vague and abstract, but this is in theory. In practice, everything becomes extremely simple. For example: the equation 0 * x = -53 has no solution, since there is no number x whose product with zero would give something other than zero.
Now we will look at the most basic types of equations.
1. Linear equation
An equation is called linear if its right and left sides are represented as linear functions: ax + b = cx + d or in generalized form kx + b = 0. Where a, b, c, d are known numbers, and x is an unknown quantity . Which equation has no roots? Examples linear equations are presented in the illustration below.
Basically, linear equations are solved by simply transferring the number part to one part and the contents of x to another. The result is an equation of the form mx = n, where m and n are numbers, and x is an unknown. To find x, just divide both sides by m. Then x = n/m. Most linear equations have only one root, but there are cases when there are either infinitely many roots or no roots at all. When m = 0 and n = 0, the equation takes the form 0 * x = 0. The solution to such an equation will be absolutely any number.
However, what equation has no roots?
For m = 0 and n = 0, the equation has no roots in the set of real numbers. 0 * x = -1; 0 * x = 200 - these equations have no roots.
2. Quadratic equation
A quadratic equation is an equation of the form ax 2 + bx + c = 0 for a = 0. The most common solution is through the discriminant. The formula for finding the discriminant of a quadratic equation is: D = b 2 - 4 * a * c. Next there are two roots x 1.2 = (-b ± √D) / 2 * a.
For D > 0 the equation has two roots, for D = 0 it has one root. But what quadratic equation has no roots? The easiest way to observe the number of roots of a quadratic equation is by graphing the function, which is a parabola. For a > 0 the branches are directed upward, for a< 0 ветви опущены вниз. Если дискриминант отрицателен, такое квадратное уравнение не имеет корней на множестве действительных чисел.
You can also visually determine the number of roots without calculating the discriminant. To do this, you need to find the vertex of the parabola and determine in which direction the branches are directed. The x coordinate of the vertex can be determined using the formula: x 0 = -b / 2a. In this case, the y coordinate of the vertex is found by simply substituting the x 0 value into the original equation.
The quadratic equation x 2 - 8x + 72 = 0 has no roots, since it has a negative discriminant D = (-8) 2 - 4 * 1 * 72 = -224. This means that the parabola does not touch the x-axis and the function never takes the value 0, therefore, the equation has no real roots.
3. Trigonometric equations
Trigonometric functions are considered on a trigonometric circle, but can also be represented in a Cartesian coordinate system. In this article we will look at two main trigonometric functions and their equations: sinx and cosx. Since these functions form a trigonometric circle with radius 1, |sinx| and |cosx| cannot be greater than 1. So, which sinx equation has no roots? Consider the graph of the sinx function shown in the picture below.
We see that the function is symmetric and has a repetition period of 2pi. Based on this, we can say that the maximum value of this function can be 1, and the minimum -1. For example, the expression cosx = 5 will not have roots, since its absolute value is greater than one.
This is the simplest example of trigonometric equations. In fact, solving them can take many pages, at the end of which you realize that you used the wrong formula and need to start all over again. Sometimes, even if you find the roots correctly, you may forget to take into account the restrictions on OD, which is why an extra root or interval appears in the answer, and the entire answer turns into an error. Therefore, strictly follow all the restrictions, because not all roots fit into the scope of the task.
4. Systems of equations
A system of equations is a set of equations joined by curly or square brackets. The curly brackets indicate that all equations are run together. That is, if at least one of the equations does not have roots or contradicts another, the entire system has no solution. Square brackets indicate the word "or". This means that if at least one of the equations of the system has a solution, then the entire system has a solution.
The answer of the system c is the set of all the roots of the individual equations. And systems with curly braces have only common roots. Systems of equations can include completely different functions, so such complexity does not allow us to immediately say which equation does not have roots.
In problem books and textbooks there are different types of equations: those that have roots and those that do not. First of all, if you can’t find the roots, don’t think that they are not there at all. Perhaps you made a mistake somewhere, then you just need to carefully double-check your decision.
We looked at the most basic equations and their types. Now you can tell which equation has no roots. In most cases this is not difficult to do. Achieving success in solving equations requires only attention and concentration. Practice more, it will help you navigate the material much better and faster.
So, the equation has no roots if:
- in the linear equation mx = n the value is m = 0 and n = 0;
- in a quadratic equation, if the discriminant is less than zero;
- in a trigonometric equation of the form cosx = m / sinx = n, if |m| > 0, |n| > 0;
- in a system of equations with curly brackets if at least one equation has no roots, and with square brackets if all equations have no roots.
", that is, equations of the first degree. In this lesson we will look at what is called a quadratic equation and how to solve it.
What is a quadratic equation?
Important!
The degree of an equation is determined by the highest degree to which the unknown stands.
If the maximum power in which the unknown is “2”, then you have a quadratic equation.
Examples of quadratic equations
- 5x 2 − 14x + 17 = 0
- −x 2 + x +
= 01 3 - x 2 + 0.25x = 0
- x 2 − 8 = 0
Important! The general form of a quadratic equation looks like this:
A x 2 + b x + c = 0
“a”, “b” and “c” are given numbers.- “a” is the first or highest coefficient;
- “b” is the second coefficient;
- “c” is a free member.
To find “a”, “b” and “c” you need to compare your equation with the general form of the quadratic equation “ax 2 + bx + c = 0”.
Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.
| The equation | Odds | |||
|---|---|---|---|---|
|
||||
|
||||
| 1 |
| 3 |
- a = −1
- b = 1
- c =
1 3
- a = 1
- b = 0.25
- c = 0
- a = 1
- b = 0
- c = −8
How to Solve Quadratic Equations
Unlike linear equations, a special method is used to solve quadratic equations. formula for finding roots.
Remember!
To solve a quadratic equation you need:
- bring the quadratic equation to the general form “ax 2 + bx + c = 0”. That is, only “0” should remain on the right side;
- use formula for roots:
Let's look at an example of how to use the formula to find the roots of a quadratic equation. Let's solve a quadratic equation.
X 2 − 3x − 4 = 0
The equation “x 2 − 3x − 4 = 0” has already been reduced to the general form “ax 2 + bx + c = 0” and does not require additional simplifications. To solve it, we just need to apply formula for finding the roots of a quadratic equation.
Let us determine the coefficients “a”, “b” and “c” for this equation.
x 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
It can be used to solve any quadratic equation.
In the formula “x 1;2 = ” the radical expression is often replaced
“b 2 − 4ac” for the letter “D” and is called discriminant. The concept of a discriminant is discussed in more detail in the lesson “What is a discriminant”.
Let's look at another example of a quadratic equation.
x 2 + 9 + x = 7x
In this form, it is quite difficult to determine the coefficients “a”, “b” and “c”. Let's first reduce the equation to the general form “ax 2 + bx + c = 0”.
X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x 2 + 9 − 6x = 0
x 2 − 6x + 9 = 0
Now you can use the formula for the roots.
X 1;2 =
x 1;2 =
x 1;2 =
x 1;2 =
x =
| 6 |
| 2 |
x = 3
Answer: x = 3
There are times when quadratic equations have no roots. This situation occurs when the formula contains a negative number under the root.
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