Solve x^2-18x-135=0

$${ x }^{ 2 } -18x-135 = 0$$

Solve for x

$$x=6\sqrt{6}+9\approx 23.696938457$$

$$x=9-6\sqrt{6}\approx -5.696938457$$

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.

$$x^{2}-18x-135=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and -135 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-135\right)}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-135\right)}}{2}$$

Multiply -4 times -135.

$$x=\frac{-\left(-18\right)±\sqrt{324+540}}{2}$$

Add 324 to 540.

$$x=\frac{-\left(-18\right)±\sqrt{864}}{2}$$

Take the square root of 864.

$$x=\frac{-\left(-18\right)±12\sqrt{6}}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±12\sqrt{6}}{2}$$

Now solve the equation x=\frac{18±12\sqrt{6}}{2} when ± is plus. Add 18 to 12\sqrt{6}.

$$x=\frac{12\sqrt{6}+18}{2}$$

Divide 18+12\sqrt{6} by 2.

$$x=6\sqrt{6}+9$$

Now solve the equation x=\frac{18±12\sqrt{6}}{2} when ± is minus. Subtract 12\sqrt{6} from 18.

$$x=\frac{18-12\sqrt{6}}{2}$$

Divide 18-12\sqrt{6} by 2.

$$x=9-6\sqrt{6}$$

The equation is now solved.

$$x=6\sqrt{6}+9 x=9-6\sqrt{6}$$

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.

$$x^{2}-18x-135=0$$

Add 135 to both sides of the equation.

$$x^{2}-18x-135-\left(-135\right)=-\left(-135\right)$$

Subtracting -135 from itself leaves 0.

$$x^{2}-18x=-\left(-135\right)$$

Subtract -135 from 0.

$$x^{2}-18x=135$$

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.

$$x^{2}-18x+\left(-9\right)^{2}=135+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=135+81$$

Add 135 to 81.

$$x^{2}-18x+81=216$$

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}.

$$\left(x-9\right)^{2}=216$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{216}$$

Simplify.

$$x-9=6\sqrt{6} x-9=-6\sqrt{6}$$

Add 9 to both sides of the equation.

$$x=6\sqrt{6}+9 x=9-6\sqrt{6}$$