Solve x^2+36x+81=0

$${ x }^{ 2 } +36x+81 = 0$$

Solve for x

$$x=9\sqrt{3}-18\approx -2.411542732$$

$$x=-9\sqrt{3}-18\approx -33.588457268$$

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.

$$x^{2}+36x+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 36 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 81}}{2}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 81}}{2}$$

Multiply -4 times 81.

$$x=\frac{-36±\sqrt{1296-324}}{2}$$

Add 1296 to -324.

$$x=\frac{-36±\sqrt{972}}{2}$$

Take the square root of 972.

$$x=\frac{-36±18\sqrt{3}}{2}$$

Now solve the equation x=\frac{-36±18\sqrt{3}}{2} when ± is plus. Add -36 to 18\sqrt{3}.

$$x=\frac{18\sqrt{3}-36}{2}$$

Divide -36+18\sqrt{3} by 2.

$$x=9\sqrt{3}-18$$

Now solve the equation x=\frac{-36±18\sqrt{3}}{2} when ± is minus. Subtract 18\sqrt{3} from -36.

$$x=\frac{-18\sqrt{3}-36}{2}$$

Divide -36-18\sqrt{3} by 2.

$$x=-9\sqrt{3}-18$$

The equation is now solved.

$$x=9\sqrt{3}-18 x=-9\sqrt{3}-18$$

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.

$$x^{2}+36x+81=0$$

Subtract 81 from both sides of the equation.

$$x^{2}+36x+81-81=-81$$

Subtracting 81 from itself leaves 0.

$$x^{2}+36x=-81$$

Divide 36, the coefficient of the x term, by 2 to get 18. Then add the square of 18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+36x+18^{2}=-81+18^{2}$$

Square 18.

$$x^{2}+36x+324=-81+324$$

Add -81 to 324.

$$x^{2}+36x+324=243$$

Factor x^{2}+36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+18\right)^{2}=243$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+18\right)^{2}}=\sqrt{243}$$

Simplify.

$$x+18=9\sqrt{3} x+18=-9\sqrt{3}$$

Subtract 18 from both sides of the equation.

$$x=9\sqrt{3}-18 x=-9\sqrt{3}-18$$