Solve x^2+18x-208

$${ x }^{ 2 } +18x-208$$

Factor

$$\left(x-8\right)\left(x+26\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-208. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\left(-208\right)=-208$$

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 -208.

$$-1,208 -2,104 -4,52 -8,26 -13,16$$

Calculate the sum for each pair.

$$-1+208=207 -2+104=102 -4+52=48 -8+26=18 -13+16=3$$

The solution is the pair that gives sum 18.

$$a=-8 b=26$$

Rewrite x^{2}+18x-208 as \left(x^{2}-8x\right)+\left(26x-208\right).

$$\left(x^{2}-8x\right)+\left(26x-208\right)$$

Factor out x in the first and 26 in the second group.

$$x\left(x-8\right)+26\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+26\right)$$

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.

$$x^{2}+18x-208=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-208\right)}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-208\right)}}{2}$$

Multiply -4 times -208.

$$x=\frac{-18±\sqrt{324+832}}{2}$$

Add 324 to 832.

$$x=\frac{-18±\sqrt{1156}}{2}$$

Take the square root of 1156.

$$x=\frac{-18±34}{2}$$

Now solve the equation x=\frac{-18±34}{2} when ± is plus. Add -18 to 34.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-18±34}{2} when ± is minus. Subtract 34 from -18.

$$x=-\frac{52}{2}$$

Divide -52 by 2.

$$x=-26$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -26 for x_{2}.

$$x^{2}+18x-208=\left(x-8\right)\left(x-\left(-26\right)\right)$$

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

$$x^{2}+18x-208=\left(x-8\right)\left(x+26\right)$$

Evaluate

$$\left(x-8\right)\left(x+26\right)$$