Solve 0=x^2+12x-18

$$0 = x ^ { 2 } + 12 x - 18$$

Solve for x

$$x=3\sqrt{6}-6\approx 1.348469228$$

$$x=-3\sqrt{6}-6\approx -13.348469228$$

Steps Using the Quadratic Formula

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

$$x^{2}+12x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-18\right)}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-12±\sqrt{144+72}}{2}$$

Add 144 to 72.

$$x=\frac{-12±\sqrt{216}}{2}$$

Take the square root of 216.

$$x=\frac{-12±6\sqrt{6}}{2}$$

Now solve the equation x=\frac{-12±6\sqrt{6}}{2} when ± is plus. Add -12 to 6\sqrt{6}.

$$x=\frac{6\sqrt{6}-12}{2}$$

Divide -12+6\sqrt{6} by 2.

$$x=3\sqrt{6}-6$$

Now solve the equation x=\frac{-12±6\sqrt{6}}{2} when ± is minus. Subtract 6\sqrt{6} from -12.

$$x=\frac{-6\sqrt{6}-12}{2}$$

Divide -12-6\sqrt{6} by 2.

$$x=-3\sqrt{6}-6$$

The equation is now solved.

$$x=3\sqrt{6}-6 x=-3\sqrt{6}-6$$

Steps for Completing the Square

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

$$x^{2}+12x-18=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$x^{2}+12x=18$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=18+6^{2}$$

Square 6.

$$x^{2}+12x+36=18+36$$

Add 18 to 36.

$$x^{2}+12x+36=54$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=54$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{54}$$

Simplify.

$$x+6=3\sqrt{6} x+6=-3\sqrt{6}$$

Subtract 6 from both sides of the equation.

$$x=3\sqrt{6}-6 x=-3\sqrt{6}-6$$