Solve x^2+16x+78=7

$$x ^ { 2 } + 16 x + 78 = 7$$

Solve for x (complex solution)

$$x=-8+\sqrt{7}i\approx -8+2.645751311i$$

$$x=-\sqrt{7}i-8\approx -8-2.645751311i$$

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.

$$x^{2}+16x+78=7$$

Subtract 7 from both sides of the equation.

$$x^{2}+16x+78-7=7-7$$

Subtracting 7 from itself leaves 0.

$$x^{2}+16x+78-7=0$$

Subtract 7 from 78.

$$x^{2}+16x+71=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 71 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 71}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 71}}{2}$$

Multiply -4 times 71.

$$x=\frac{-16±\sqrt{256-284}}{2}$$

Add 256 to -284.

$$x=\frac{-16±\sqrt{-28}}{2}$$

Take the square root of -28.

$$x=\frac{-16±2\sqrt{7}i}{2}$$

Now solve the equation x=\frac{-16±2\sqrt{7}i}{2} when ± is plus. Add -16 to 2i\sqrt{7}.

$$x=\frac{-16+2\sqrt{7}i}{2}$$

Divide -16+2i\sqrt{7} by 2.

$$x=-8+\sqrt{7}i$$

Now solve the equation x=\frac{-16±2\sqrt{7}i}{2} when ± is minus. Subtract 2i\sqrt{7} from -16.

$$x=\frac{-2\sqrt{7}i-16}{2}$$

Divide -16-2i\sqrt{7} by 2.

$$x=-\sqrt{7}i-8$$

The equation is now solved.

$$x=-8+\sqrt{7}i x=-\sqrt{7}i-8$$

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.

$$x^{2}+16x+78=7$$

Subtract 78 from both sides of the equation.

$$x^{2}+16x+78-78=7-78$$

Subtracting 78 from itself leaves 0.

$$x^{2}+16x=7-78$$

Subtract 78 from 7.

$$x^{2}+16x=-71$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=-71+8^{2}$$

Square 8.

$$x^{2}+16x+64=-71+64$$

Add -71 to 64.

$$x^{2}+16x+64=-7$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=-7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{-7}$$

Simplify.

$$x+8=\sqrt{7}i x+8=-\sqrt{7}i$$

Subtract 8 from both sides of the equation.

$$x=-8+\sqrt{7}i x=-\sqrt{7}i-8$$