Solve 8x^2=-40-16x

$$8 x ^ { 2 } = - 40 - 16 x$$

Solve for x (complex solution)

$$x=-1+2i$$

$$x=-1-2i$$

Steps Using the Quadratic Formula

Subtract -40 from both sides.

$$8x^{2}-\left(-40\right)=-16x$$

The opposite of -40 is 40.

$$8x^{2}+40=-16x$$

Add 16x to both sides.

$$8x^{2}+40+16x=0$$

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.

$$8x^{2}+16x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 16 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 8\times 40}}{2\times 8}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 8\times 40}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-16±\sqrt{256-32\times 40}}{2\times 8}$$

Multiply -32 times 40.

$$x=\frac{-16±\sqrt{256-1280}}{2\times 8}$$

Add 256 to -1280.

$$x=\frac{-16±\sqrt{-1024}}{2\times 8}$$

Take the square root of -1024.

$$x=\frac{-16±32i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-16±32i}{16}$$

Now solve the equation x=\frac{-16±32i}{16} when ± is plus. Add -16 to 32i.

$$x=\frac{-16+32i}{16}$$

Divide -16+32i by 16.

$$x=-1+2i$$

Now solve the equation x=\frac{-16±32i}{16} when ± is minus. Subtract 32i from -16.

$$x=\frac{-16-32i}{16}$$

Divide -16-32i by 16.

$$x=-1-2i$$

The equation is now solved.

$$x=-1+2i x=-1-2i$$

Steps for Completing the Square

Add 16x to both sides.

$$8x^{2}+16x=-40$$

Divide both sides by 8.

$$\frac{8x^{2}+16x}{8}=-\frac{40}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{16}{8}x=-\frac{40}{8}$$

Divide 16 by 8.

$$x^{2}+2x=-\frac{40}{8}$$

Divide -40 by 8.

$$x^{2}+2x=-5$$

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.

$$x^{2}+2x+1^{2}=-5+1^{2}$$

Square 1.

$$x^{2}+2x+1=-5+1$$

Add -5 to 1.

$$x^{2}+2x+1=-4$$

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}.

$$\left(x+1\right)^{2}=-4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-4}$$

Simplify.

$$x+1=2i x+1=-2i$$

Subtract 1 from both sides of the equation.

$$x=-1+2i x=-1-2i$$