Solve x^2+18x+117=0

$${ x }^{ 2 } +18x+117=0$$

Solve for x (complex solution)

$$x=-9+6i$$

$$x=-9-6i$$

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.

$$x^{2}+18x+117=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and 117 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 117}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 117}}{2}$$

Multiply -4 times 117.

$$x=\frac{-18±\sqrt{324-468}}{2}$$

Add 324 to -468.

$$x=\frac{-18±\sqrt{-144}}{2}$$

Take the square root of -144.

$$x=\frac{-18±12i}{2}$$

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

$$x=\frac{-18+12i}{2}$$

Divide -18+12i by 2.

$$x=-9+6i$$

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

$$x=\frac{-18-12i}{2}$$

Divide -18-12i by 2.

$$x=-9-6i$$

The equation is now solved.

$$x=-9+6i x=-9-6i$$

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.

$$x^{2}+18x+117=0$$

Subtract 117 from both sides of the equation.

$$x^{2}+18x+117-117=-117$$

Subtracting 117 from itself leaves 0.

$$x^{2}+18x=-117$$

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.

$$x^{2}+18x+9^{2}=-117+9^{2}$$

Square 9.

$$x^{2}+18x+81=-117+81$$

Add -117 to 81.

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

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}.

$$\left(x+9\right)^{2}=-36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{-36}$$

Simplify.

$$x+9=6i x+9=-6i$$

Subtract 9 from both sides of the equation.

$$x=-9+6i x=-9-6i$$