Solve 5x^2=7x-6

$$5 x ^ { 2 } = 7 x - 6$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{71}i}{10}\approx 0.7+0.842614977i$$

$$x=\frac{-\sqrt{71}i+7}{10}\approx 0.7-0.842614977i$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$5x^{2}-7x=-6$$

Add 6 to both sides.

$$5x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 5\times 6}}{2\times 5}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 5\times 6}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-7\right)±\sqrt{49-20\times 6}}{2\times 5}$$

Multiply -20 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-120}}{2\times 5}$$

Add 49 to -120.

$$x=\frac{-\left(-7\right)±\sqrt{-71}}{2\times 5}$$

Take the square root of -71.

$$x=\frac{-\left(-7\right)±\sqrt{71}i}{2\times 5}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{71}i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{7±\sqrt{71}i}{10}$$

Now solve the equation x=\frac{7±\sqrt{71}i}{10} when ± is plus. Add 7 to i\sqrt{71}.

$$x=\frac{7+\sqrt{71}i}{10}$$

Now solve the equation x=\frac{7±\sqrt{71}i}{10} when ± is minus. Subtract i\sqrt{71} from 7.

$$x=\frac{-\sqrt{71}i+7}{10}$$

The equation is now solved.

$$x=\frac{7+\sqrt{71}i}{10} x=\frac{-\sqrt{71}i+7}{10}$$

Steps for Completing the Square

Subtract 7x from both sides.

$$5x^{2}-7x=-6$$

Divide both sides by 5.

$$\frac{5x^{2}-7x}{5}=-\frac{6}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{7}{5}x=-\frac{6}{5}$$

Divide -\frac{7}{5}, the coefficient of the x term, by 2 to get -\frac{7}{10}. Then add the square of -\frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=-\frac{6}{5}+\left(-\frac{7}{10}\right)^{2}$$

Square -\frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=-\frac{6}{5}+\frac{49}{100}$$

Add -\frac{6}{5} to \frac{49}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=-\frac{71}{100}$$

Factor x^{2}-\frac{7}{5}x+\frac{49}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{10}\right)^{2}=-\frac{71}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{-\frac{71}{100}}$$

Simplify.

$$x-\frac{7}{10}=\frac{\sqrt{71}i}{10} x-\frac{7}{10}=-\frac{\sqrt{71}i}{10}$$

Add \frac{7}{10} to both sides of the equation.

$$x=\frac{7+\sqrt{71}i}{10} x=\frac{-\sqrt{71}i+7}{10}$$