How to factorize a square trinomial: formula
Expanding polynomials to get a product sometimes seems confusing. But it is not so difficult if you understand the process step by step. The article details how to factorize a square trinomial.
Many do not understand how to factorize a square trinomial, and why this is done. At first it may seem that this is a useless exercise. But in mathematics, nothing is done just like that. The transformation is necessary to simplify the expression and the convenience of calculation.
A polynomial having the form - ax² + bx + c, is called a square trinomial. The term "a" must be negative or positive. In practice, this expression is called a quadratic equation. Therefore, sometimes they say differently: how to expand a quadratic equation.
Interesting! A square polynomial is called because of its largest degree - a square. And a trinomial - because of the 3 component terms.
Some other kinds of polynomials:
- linear binomial (6x+8);
- cubic quadrilateral (x³+4x²-2x+9).
Factorization of a square trinomial
First, the expression is equal to zero, then you need to find the values of the roots x1 and x2. There may be no roots, there may be one or two roots. The presence of roots is determined by the discriminant. Its formula must be known by heart: D=b²-4ac.
If the result of D is negative, there are no roots. If positive, there are two roots. If the result is zero, the root is one. The roots are also calculated by the formula.
If the calculation of the discriminant results in zero, you can apply any of the formulas. In practice, the formula is simply abbreviated: -b / 2a.
Formulas for different values of the discriminant are different.
If D is positive:
If D is zero:
Online calculators
There is an online calculator on the Internet. It can be used to factorize. Some resources provide the opportunity to see the solution step by step. Such services help to better understand the topic, but you need to try to understand well.
Useful video: Factoring a square trinomial
Examples
We suggest looking at simple examples of how to factorize a quadratic equation.
Example 1
Here it is clearly shown that the result will be two x, because D is positive. They need to be substituted into the formula. If the roots are negative, the sign in the formula is reversed.
We know the formula for factoring a square trinomial: a(x-x1)(x-x2). We put the values in brackets: (x+3)(x+2/3). There is no number before the term in the exponent. This means that there is a unit, it is lowered.
Example 2
This example clearly shows how to solve an equation that has one root.
Substitute the resulting value:
Example 3
Given: 5x²+3x+7
First, we calculate the discriminant, as in the previous cases.
D=9-4*5*7=9-140= -131.
The discriminant is negative, which means there are no roots.
After receiving the result, it is worth opening the brackets and checking the result. The original trinomial should appear.
Alternative solution
Some people have never been able to make friends with the discriminant. There is another way to factorize a square trinomial. For convenience, the method is shown in an example.
Given: x²+3x-10
We know that we should end up with 2 parentheses: (_)(_). When the expression looks like this: x² + bx + c, we put x at the beginning of each bracket: (x_) (x_). The remaining two numbers are the product that gives "c", i.e. -10 in this case. To find out what these numbers are, you can only use the selection method. Substituted numbers must match the remaining term.
For example, multiplying the following numbers gives -10:
- -1, 10;
- -10, 1;
- -5, 2;
- -2, 5.
- (x-1)(x+10) = x2+10x-x-10 = x2+9x-10. No.
- (x-10)(x+1) = x2+x-10x-10 = x2-9x-10. No.
- (x-5)(x+2) = x2+2x-5x-10 = x2-3x-10. No.
- (x-2)(x+5) = x2+5x-2x-10 = x2+3x-10. Fits.
So, the transformation of the expression x2+3x-10 looks like this: (x-2)(x+5).
Important! You should be careful not to confuse the signs.
Decomposition of a complex trinomial
If "a" is greater than one, difficulties begin. But everything is not as difficult as it seems.
In order to factorize, one must first see if it is possible to factor something out.
For example, given the expression: 3x²+9x-30. Here the number 3 is taken out of brackets:
3(x²+3x-10). The result is the already known trinomial. The answer looks like this: 3(x-2)(x+5)
How to decompose if the term that is squared is negative? In this case, the number -1 is taken out of the bracket. For example: -x²-10x-8. The expression will then look like this:
The scheme differs little from the previous one. There are only a few new things. Let's say the expression is given: 2x²+7x+3. The answer is also written in 2 brackets, which must be filled in (_) (_). X is written in the 2nd bracket, and what is left in the 1st. It looks like this: (2x_)(x_). Otherwise, the previous scheme is repeated.
The number 3 gives the numbers:
- -1, -3;
- -3, -1;
- 3, 1;
- 1, 3.
We solve equations by substituting the given numbers. The last option fits. So the transformation of the expression 2x²+7x+3 looks like this: (2x+1)(x+3).
Other cases
It is not always possible to transform an expression. In the second method, the solution of the equation is not required. But the possibility of converting terms into a product is checked only through the discriminant.
It is worth practicing solving quadratic equations so that there are no difficulties when using formulas.
Useful video: factorization of a trinomial
Conclusion
You can use it in any way. But it is better to work both to automatism. Also, those who are going to connect their lives with mathematics need to learn how to solve quadratic equations well and decompose polynomials into factors. All the following mathematical topics are built on this.
