Solve 7x^2=11x-10

$$7 x ^ { 2 } = 11 x - 10$$

Solve for x (complex solution)

$$x=\frac{11+\sqrt{159}i}{14}\approx 0.785714286+0.900680015i$$

$$x=\frac{-\sqrt{159}i+11}{14}\approx 0.785714286-0.900680015i$$

Steps Using the Quadratic Formula

Subtract 11x from both sides.

$$7x^{2}-11x=-10$$

Add 10 to both sides.

$$7x^{2}-11x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -11 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 7\times 10}}{2\times 7}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 7\times 10}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-11\right)±\sqrt{121-28\times 10}}{2\times 7}$$

Multiply -28 times 10.

$$x=\frac{-\left(-11\right)±\sqrt{121-280}}{2\times 7}$$

Add 121 to -280.

$$x=\frac{-\left(-11\right)±\sqrt{-159}}{2\times 7}$$

Take the square root of -159.

$$x=\frac{-\left(-11\right)±\sqrt{159}i}{2\times 7}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{159}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{11±\sqrt{159}i}{14}$$

Now solve the equation x=\frac{11±\sqrt{159}i}{14} when ± is plus. Add 11 to i\sqrt{159}.

$$x=\frac{11+\sqrt{159}i}{14}$$

Now solve the equation x=\frac{11±\sqrt{159}i}{14} when ± is minus. Subtract i\sqrt{159} from 11.

$$x=\frac{-\sqrt{159}i+11}{14}$$

The equation is now solved.

$$x=\frac{11+\sqrt{159}i}{14} x=\frac{-\sqrt{159}i+11}{14}$$

Steps for Completing the Square

Subtract 11x from both sides.

$$7x^{2}-11x=-10$$

Divide both sides by 7.

$$\frac{7x^{2}-11x}{7}=-\frac{10}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{11}{7}x=-\frac{10}{7}$$

Divide -\frac{11}{7}, the coefficient of the x term, by 2 to get -\frac{11}{14}. Then add the square of -\frac{11}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{7}x+\left(-\frac{11}{14}\right)^{2}=-\frac{10}{7}+\left(-\frac{11}{14}\right)^{2}$$

Square -\frac{11}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{7}x+\frac{121}{196}=-\frac{10}{7}+\frac{121}{196}$$

Add -\frac{10}{7} to \frac{121}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{7}x+\frac{121}{196}=-\frac{159}{196}$$

Factor x^{2}-\frac{11}{7}x+\frac{121}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{14}\right)^{2}=-\frac{159}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{14}\right)^{2}}=\sqrt{-\frac{159}{196}}$$

Simplify.

$$x-\frac{11}{14}=\frac{\sqrt{159}i}{14} x-\frac{11}{14}=-\frac{\sqrt{159}i}{14}$$

Add \frac{11}{14} to both sides of the equation.

$$x=\frac{11+\sqrt{159}i}{14} x=\frac{-\sqrt{159}i+11}{14}$$