Solve x^2+31x-816

$${ x }^{ 2 } +31x-816$$

Factor

$$\left(x-17\right)\left(x+48\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-816. To find a and b, set up a system to be solved.

$$a+b=31 ab=1\left(-816\right)=-816$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -816.

$$-1,816 -2,408 -3,272 -4,204 -6,136 -8,102 -12,68 -16,51 -17,48 -24,34$$

Calculate the sum for each pair.

$$-1+816=815 -2+408=406 -3+272=269 -4+204=200 -6+136=130 -8+102=94 -12+68=56 -16+51=35 -17+48=31 -24+34=10$$

The solution is the pair that gives sum 31.

$$a=-17 b=48$$

Rewrite x^{2}+31x-816 as \left(x^{2}-17x\right)+\left(48x-816\right).

$$\left(x^{2}-17x\right)+\left(48x-816\right)$$

Factor out x in the first and 48 in the second group.

$$x\left(x-17\right)+48\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x+48\right)$$

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.

$$x^{2}+31x-816=0$$

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.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-816\right)}}{2}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\left(-816\right)}}{2}$$

Multiply -4 times -816.

$$x=\frac{-31±\sqrt{961+3264}}{2}$$

Add 961 to 3264.

$$x=\frac{-31±\sqrt{4225}}{2}$$

Take the square root of 4225.

$$x=\frac{-31±65}{2}$$

Now solve the equation x=\frac{-31±65}{2} when ± is plus. Add -31 to 65.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{-31±65}{2} when ± is minus. Subtract 65 from -31.

$$x=-\frac{96}{2}$$

Divide -96 by 2.

$$x=-48$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17 for x_{1} and -48 for x_{2}.

$$x^{2}+31x-816=\left(x-17\right)\left(x-\left(-48\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+31x-816=\left(x-17\right)\left(x+48\right)$$

Evaluate

$$\left(x-17\right)\left(x+48\right)$$