Solve x^2-18x=81

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

Solve for x

$$x=9\sqrt{2}+9\approx 21.727922061$$

$$x=9-9\sqrt{2}\approx -3.727922061$$

Steps Using the Quadratic Formula

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.

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

Subtract 81 from both sides of the equation.

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

Subtracting 81 from itself leaves 0.

$$x^{2}-18x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-81\right)}}{2}$$

Multiply -4 times -81.

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

Add 324 to 324.

$$x=\frac{-\left(-18\right)±\sqrt{648}}{2}$$

Take the square root of 648.

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

The opposite of -18 is 18.

$$x=\frac{18±18\sqrt{2}}{2}$$

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

$$x=\frac{18\sqrt{2}+18}{2}$$

Divide 18+18\sqrt{2} by 2.

$$x=9\sqrt{2}+9$$

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

$$x=\frac{18-18\sqrt{2}}{2}$$

Divide 18-18\sqrt{2} by 2.

$$x=9-9\sqrt{2}$$

The equation is now solved.

$$x=9\sqrt{2}+9 x=9-9\sqrt{2}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

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.

$$x^{2}-18x+\left(-9\right)^{2}=81+\left(-9\right)^{2}$$

Square -9.

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

Add 81 to 81.

$$x^{2}-18x+81=162$$

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}.

$$\left(x-9\right)^{2}=162$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{162}$$

Simplify.

$$x-9=9\sqrt{2} x-9=-9\sqrt{2}$$

Add 9 to both sides of the equation.

$$x=9\sqrt{2}+9 x=9-9\sqrt{2}$$