Solve x^2+66x+93=0

$${ x }^{ 2 } +66x+93=0$$

Solve for x

$$x=2\sqrt{249}-33\approx -1.440532324$$

$$x=-2\sqrt{249}-33\approx -64.559467676$$

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.

$$x^{2}+66x+93=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 66 for b, and 93 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-66±\sqrt{66^{2}-4\times 93}}{2}$$

Square 66.

$$x=\frac{-66±\sqrt{4356-4\times 93}}{2}$$

Multiply -4 times 93.

$$x=\frac{-66±\sqrt{4356-372}}{2}$$

Add 4356 to -372.

$$x=\frac{-66±\sqrt{3984}}{2}$$

Take the square root of 3984.

$$x=\frac{-66±4\sqrt{249}}{2}$$

Now solve the equation x=\frac{-66±4\sqrt{249}}{2} when ± is plus. Add -66 to 4\sqrt{249}.

$$x=\frac{4\sqrt{249}-66}{2}$$

Divide -66+4\sqrt{249} by 2.

$$x=2\sqrt{249}-33$$

Now solve the equation x=\frac{-66±4\sqrt{249}}{2} when ± is minus. Subtract 4\sqrt{249} from -66.

$$x=\frac{-4\sqrt{249}-66}{2}$$

Divide -66-4\sqrt{249} by 2.

$$x=-2\sqrt{249}-33$$

The equation is now solved.

$$x=2\sqrt{249}-33 x=-2\sqrt{249}-33$$

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.

$$x^{2}+66x+93=0$$

Subtract 93 from both sides of the equation.

$$x^{2}+66x+93-93=-93$$

Subtracting 93 from itself leaves 0.

$$x^{2}+66x=-93$$

Divide 66, the coefficient of the x term, by 2 to get 33. Then add the square of 33 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+66x+33^{2}=-93+33^{2}$$

Square 33.

$$x^{2}+66x+1089=-93+1089$$

Add -93 to 1089.

$$x^{2}+66x+1089=996$$

Factor x^{2}+66x+1089. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+33\right)^{2}=996$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+33\right)^{2}}=\sqrt{996}$$

Simplify.

$$x+33=2\sqrt{249} x+33=-2\sqrt{249}$$

Subtract 33 from both sides of the equation.

$$x=2\sqrt{249}-33 x=-2\sqrt{249}-33$$