Solve 0=x^2+18x+32

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

Solve for x

$$x=-16$$

$$x=-2$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

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

To solve the equation, factor x^{2}+18x+32 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=18 ab=32$$

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 factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

To find equation solutions, solve x+2=0 and x+16=0.

$$x=-2 x=-16$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

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

To solve the equation, factor the left hand side by grouping. First, left hand side 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.

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

To find equation solutions, solve x+2=0 and x+16=0.

$$x=-2 x=-16$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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.

$$x=-16$$

The equation is now solved.

$$x=-2 x=-16$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

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

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

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

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=-32+9^{2}$$

Square 9.

$$x^{2}+18x+81=-32+81$$

Add -32 to 81.

$$x^{2}+18x+81=49$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{49}$$

Simplify.

$$x+9=7 x+9=-7$$

Subtract 9 from both sides of the equation.

$$x=-2 x=-16$$