Solve x^2-20x+91

$${ x }^{ 2 } -20x+91$$

Factor

$$\left(x-13\right)\left(x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+91. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\times 91=91$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 91.

$$-1,-91 -7,-13$$

Calculate the sum for each pair.

$$-1-91=-92 -7-13=-20$$

The solution is the pair that gives sum -20.

$$a=-13 b=-7$$

Rewrite x^{2}-20x+91 as \left(x^{2}-13x\right)+\left(-7x+91\right).

$$\left(x^{2}-13x\right)+\left(-7x+91\right)$$

Factor out x in the first and -7 in the second group.

$$x\left(x-13\right)-7\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x-7\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-20x+91=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 91}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 91}}{2}$$

Multiply -4 times 91.

$$x=\frac{-\left(-20\right)±\sqrt{400-364}}{2}$$

Add 400 to -364.

$$x=\frac{-\left(-20\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-\left(-20\right)±6}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±6}{2}$$

Now solve the equation x=\frac{20±6}{2} when ± is plus. Add 20 to 6.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{20±6}{2} when ± is minus. Subtract 6 from 20.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and 7 for x_{2}.

$$x^{2}-20x+91=\left(x-13\right)\left(x-7\right)$$

Evaluate

$$\left(x-13\right)\left(x-7\right)$$