Solve x^2+134=-2x-14

$$x ^ { 2 } + 134 = - 2 x - 14$$

Solve for x (complex solution)

$$x=-1+7\sqrt{3}i\approx -1+12.124355653i$$

$$x=-7\sqrt{3}i-1\approx -1-12.124355653i$$

Steps Using the Quadratic Formula

Add 2x to both sides.

$$x^{2}+134+2x=-14$$

Add 14 to both sides.

$$x^{2}+134+2x+14=0$$

Add 134 and 14 to get 148.

$$x^{2}+148+2x=0$$

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.

$$x^{2}+2x+148=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 148 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 148}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 148}}{2}$$

Multiply -4 times 148.

$$x=\frac{-2±\sqrt{4-592}}{2}$$

Add 4 to -592.

$$x=\frac{-2±\sqrt{-588}}{2}$$

Take the square root of -588.

$$x=\frac{-2±14\sqrt{3}i}{2}$$

Now solve the equation x=\frac{-2±14\sqrt{3}i}{2} when ± is plus. Add -2 to 14i\sqrt{3}.

$$x=\frac{-2+14\sqrt{3}i}{2}$$

Divide -2+14i\sqrt{3} by 2.

$$x=-1+7\sqrt{3}i$$

Now solve the equation x=\frac{-2±14\sqrt{3}i}{2} when ± is minus. Subtract 14i\sqrt{3} from -2.

$$x=\frac{-14\sqrt{3}i-2}{2}$$

Divide -2-14i\sqrt{3} by 2.

$$x=-7\sqrt{3}i-1$$

The equation is now solved.

$$x=-1+7\sqrt{3}i x=-7\sqrt{3}i-1$$

Steps for Completing the Square

Add 2x to both sides.

$$x^{2}+134+2x=-14$$

Subtract 134 from both sides.

$$x^{2}+2x=-14-134$$

Subtract 134 from -14 to get -148.

$$x^{2}+2x=-148$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=-148+1^{2}$$

Square 1.

$$x^{2}+2x+1=-148+1$$

Add -148 to 1.

$$x^{2}+2x+1=-147$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=-147$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-147}$$

Simplify.

$$x+1=7\sqrt{3}i x+1=-7\sqrt{3}i$$

Subtract 1 from both sides of the equation.

$$x=-1+7\sqrt{3}i x=-7\sqrt{3}i-1$$