Solve 8x^2=7x-2

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

Solve for x (complex solution)

$$x=\frac{7+\sqrt{15}i}{16}\approx 0.4375+0.242061459i$$

$$x=\frac{-\sqrt{15}i+7}{16}\approx 0.4375-0.242061459i$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

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

Add 2 to both sides.

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

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

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 8\times 2}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-7\right)±\sqrt{49-32\times 2}}{2\times 8}$$

Multiply -32 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-64}}{2\times 8}$$

Add 49 to -64.

$$x=\frac{-\left(-7\right)±\sqrt{-15}}{2\times 8}$$

Take the square root of -15.

$$x=\frac{-\left(-7\right)±\sqrt{15}i}{2\times 8}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{15}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{7±\sqrt{15}i}{16}$$

Now solve the equation x=\frac{7±\sqrt{15}i}{16} when ± is plus. Add 7 to i\sqrt{15}.

$$x=\frac{7+\sqrt{15}i}{16}$$

Now solve the equation x=\frac{7±\sqrt{15}i}{16} when ± is minus. Subtract i\sqrt{15} from 7.

$$x=\frac{-\sqrt{15}i+7}{16}$$

The equation is now solved.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$

Steps for Completing the Square

Subtract 7x from both sides.

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

Divide both sides by 8.

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

Dividing by 8 undoes the multiplication by 8.

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

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

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

Divide -\frac{7}{8}, the coefficient of the x term, by 2 to get -\frac{7}{16}. Then add the square of -\frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=-\frac{1}{4}+\left(-\frac{7}{16}\right)^{2}$$

Square -\frac{7}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{1}{4}+\frac{49}{256}$$

Add -\frac{1}{4} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{15}{256}$$

Factor x^{2}-\frac{7}{8}x+\frac{49}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{16}\right)^{2}=-\frac{15}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{-\frac{15}{256}}$$

Simplify.

$$x-\frac{7}{16}=\frac{\sqrt{15}i}{16} x-\frac{7}{16}=-\frac{\sqrt{15}i}{16}$$

Add \frac{7}{16} to both sides of the equation.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$