Solve 9x^2-6x+2=5x-6

$$9 { x }^{ 2 } -6x+2=5x-6$$

Solve for x (complex solution)

$$x=\frac{11+\sqrt{167}i}{18}\approx 0.611111111+0.717935999i$$

$$x=\frac{-\sqrt{167}i+11}{18}\approx 0.611111111-0.717935999i$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

$$9x^{2}-6x+2-5x=-6$$

Combine -6x and -5x to get -11x.

$$9x^{2}-11x+2=-6$$

Add 6 to both sides.

$$9x^{2}-11x+2+6=0$$

Add 2 and 6 to get 8.

$$9x^{2}-11x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -11 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 9\times 8}}{2\times 9}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 9\times 8}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-11\right)±\sqrt{121-36\times 8}}{2\times 9}$$

Multiply -36 times 8.

$$x=\frac{-\left(-11\right)±\sqrt{121-288}}{2\times 9}$$

Add 121 to -288.

$$x=\frac{-\left(-11\right)±\sqrt{-167}}{2\times 9}$$

Take the square root of -167.

$$x=\frac{-\left(-11\right)±\sqrt{167}i}{2\times 9}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{167}i}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{11±\sqrt{167}i}{18}$$

Now solve the equation x=\frac{11±\sqrt{167}i}{18} when ± is plus. Add 11 to i\sqrt{167}.

$$x=\frac{11+\sqrt{167}i}{18}$$

Now solve the equation x=\frac{11±\sqrt{167}i}{18} when ± is minus. Subtract i\sqrt{167} from 11.

$$x=\frac{-\sqrt{167}i+11}{18}$$

The equation is now solved.

$$x=\frac{11+\sqrt{167}i}{18} x=\frac{-\sqrt{167}i+11}{18}$$

Steps for Completing the Square

Subtract 5x from both sides.

$$9x^{2}-6x+2-5x=-6$$

Combine -6x and -5x to get -11x.

$$9x^{2}-11x+2=-6$$

Subtract 2 from both sides.

$$9x^{2}-11x=-6-2$$

Subtract 2 from -6 to get -8.

$$9x^{2}-11x=-8$$

Divide both sides by 9.

$$\frac{9x^{2}-11x}{9}=-\frac{8}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{11}{9}x=-\frac{8}{9}$$

Divide -\frac{11}{9}, the coefficient of the x term, by 2 to get -\frac{11}{18}. Then add the square of -\frac{11}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{9}x+\left(-\frac{11}{18}\right)^{2}=-\frac{8}{9}+\left(-\frac{11}{18}\right)^{2}$$

Square -\frac{11}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{9}x+\frac{121}{324}=-\frac{8}{9}+\frac{121}{324}$$

Add -\frac{8}{9} to \frac{121}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{9}x+\frac{121}{324}=-\frac{167}{324}$$

Factor x^{2}-\frac{11}{9}x+\frac{121}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{18}\right)^{2}=-\frac{167}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{18}\right)^{2}}=\sqrt{-\frac{167}{324}}$$

Simplify.

$$x-\frac{11}{18}=\frac{\sqrt{167}i}{18} x-\frac{11}{18}=-\frac{\sqrt{167}i}{18}$$

Add \frac{11}{18} to both sides of the equation.

$$x=\frac{11+\sqrt{167}i}{18} x=\frac{-\sqrt{167}i+11}{18}$$