Solve x^2+18x+32

$${ x }^{ 2 } +18x+32$$

Factor

$$\left(x+2\right)\left(x+16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+32. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\times 32=32$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 18.

$$a=2 b=16$$

Rewrite x^{2}+18x+32 as \left(x^{2}+2x\right)+\left(16x+32\right).

$$\left(x^{2}+2x\right)+\left(16x+32\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(x+2\right)+16\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+16\right)$$

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.

$$x^{2}+18x+32=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\times 32}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 32}}{2}$$

Multiply -4 times 32.

$$x=\frac{-18±\sqrt{324-128}}{2}$$

Add 324 to -128.

$$x=\frac{-18±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-18±14}{2}$$

Now solve the equation x=\frac{-18±14}{2} when ± is plus. Add -18 to 14.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-18±14}{2} when ± is minus. Subtract 14 from -18.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and -16 for x_{2}.

$$x^{2}+18x+32=\left(x-\left(-2\right)\right)\left(x-\left(-16\right)\right)$$

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

$$x^{2}+18x+32=\left(x+2\right)\left(x+16\right)$$

Evaluate

$$\left(x+2\right)\left(x+16\right)$$