Solve x^2=9x+18

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

Solve for x

$$x = \frac{3 \sqrt{17} + 9}{2} \approx 10.684658438$$

$$x=\frac{9-3\sqrt{17}}{2}\approx -1.684658438$$

Steps Using the Quadratic Formula

Subtract 9x from both sides.

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

Subtract 18 from both sides.

$$x^{2}-9x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-18\right)}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-\left(-9\right)±\sqrt{81+72}}{2}$$

Add 81 to 72.

$$x=\frac{-\left(-9\right)±\sqrt{153}}{2}$$

Take the square root of 153.

$$x=\frac{-\left(-9\right)±3\sqrt{17}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±3\sqrt{17}}{2}$$

Now solve the equation x=\frac{9±3\sqrt{17}}{2} when ± is plus. Add 9 to 3\sqrt{17}.

$$x=\frac{3\sqrt{17}+9}{2}$$

Now solve the equation x=\frac{9±3\sqrt{17}}{2} when ± is minus. Subtract 3\sqrt{17} from 9.

$$x=\frac{9-3\sqrt{17}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{17}+9}{2} x=\frac{9-3\sqrt{17}}{2}$$

Steps for Completing the Square

Subtract 9x from both sides.

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

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=18+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=18+\frac{81}{4}$$

Add 18 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{153}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{153}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{153}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{3\sqrt{17}}{2} x-\frac{9}{2}=-\frac{3\sqrt{17}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{3\sqrt{17}+9}{2} x=\frac{9-3\sqrt{17}}{2}$$