Solve 8x^2=-7-2x

$$8 x ^ { 2 } = - 7 - 2 x$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{55}i}{8}\approx -0.125+0.927024811i$$

$$x=\frac{-\sqrt{55}i-1}{8}\approx -0.125-0.927024811i$$

Steps Using the Quadratic Formula

Subtract -7 from both sides.

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

The opposite of -7 is 7.

$$8x^{2}+7=-2x$$

Add 2x to both sides.

$$8x^{2}+7+2x=0$$

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.

$$8x^{2}+2x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 8\times 7}}{2\times 8}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 8\times 7}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-2±\sqrt{4-32\times 7}}{2\times 8}$$

Multiply -32 times 7.

$$x=\frac{-2±\sqrt{4-224}}{2\times 8}$$

Add 4 to -224.

$$x=\frac{-2±\sqrt{-220}}{2\times 8}$$

Take the square root of -220.

$$x=\frac{-2±2\sqrt{55}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-2±2\sqrt{55}i}{16}$$

Now solve the equation x=\frac{-2±2\sqrt{55}i}{16} when ± is plus. Add -2 to 2i\sqrt{55}.

$$x=\frac{-2+2\sqrt{55}i}{16}$$

Divide -2+2i\sqrt{55} by 16.

$$x=\frac{-1+\sqrt{55}i}{8}$$

Now solve the equation x=\frac{-2±2\sqrt{55}i}{16} when ± is minus. Subtract 2i\sqrt{55} from -2.

$$x=\frac{-2\sqrt{55}i-2}{16}$$

Divide -2-2i\sqrt{55} by 16.

$$x=\frac{-\sqrt{55}i-1}{8}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{55}i}{8} x=\frac{-\sqrt{55}i-1}{8}$$

Steps for Completing the Square

Add 2x to both sides.

$$8x^{2}+2x=-7$$

Divide both sides by 8.

$$\frac{8x^{2}+2x}{8}=-\frac{7}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{2}{8}x=-\frac{7}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{4}x=-\frac{7}{8}$$

Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=-\frac{7}{8}+\left(\frac{1}{8}\right)^{2}$$

Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=-\frac{7}{8}+\frac{1}{64}$$

Add -\frac{7}{8} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=-\frac{55}{64}$$

Factor x^{2}+\frac{1}{4}x+\frac{1}{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}.

$$\left(x+\frac{1}{8}\right)^{2}=-\frac{55}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{-\frac{55}{64}}$$

Simplify.

$$x+\frac{1}{8}=\frac{\sqrt{55}i}{8} x+\frac{1}{8}=-\frac{\sqrt{55}i}{8}$$

Subtract \frac{1}{8} from both sides of the equation.

$$x=\frac{-1+\sqrt{55}i}{8} x=\frac{-\sqrt{55}i-1}{8}$$