Solve -9b^2=-8b+8

$$- 9 b ^ { 2 } = - 8 b + 8$$

Solve for b

$$b=\frac{-2\sqrt{14}i+4}{9}\approx 0.444444444-0.831479419i$$

$$b=\frac{4+2\sqrt{14}i}{9}\approx 0.444444444+0.831479419i$$

Steps Using the Quadratic Formula

Add 8b to both sides.

$$-9b^{2}+8b=8$$

Subtract 8 from both sides.

$$-9b^{2}+8b-8=0$$

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

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

Square 8.

$$b=\frac{-8±\sqrt{64-4\left(-9\right)\left(-8\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$b=\frac{-8±\sqrt{64+36\left(-8\right)}}{2\left(-9\right)}$$

Multiply 36 times -8.

$$b=\frac{-8±\sqrt{64-288}}{2\left(-9\right)}$$

Add 64 to -288.

$$b=\frac{-8±\sqrt{-224}}{2\left(-9\right)}$$

Take the square root of -224.

$$b=\frac{-8±4\sqrt{14}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$b=\frac{-8±4\sqrt{14}i}{-18}$$

Now solve the equation b=\frac{-8±4\sqrt{14}i}{-18} when ± is plus. Add -8 to 4i\sqrt{14}.

$$b=\frac{-8+4\sqrt{14}i}{-18}$$

Divide -8+4i\sqrt{14} by -18.

$$b=\frac{-2\sqrt{14}i+4}{9}$$

Now solve the equation b=\frac{-8±4\sqrt{14}i}{-18} when ± is minus. Subtract 4i\sqrt{14} from -8.

$$b=\frac{-4\sqrt{14}i-8}{-18}$$

Divide -8-4i\sqrt{14} by -18.

$$b=\frac{4+2\sqrt{14}i}{9}$$

The equation is now solved.

$$b=\frac{-2\sqrt{14}i+4}{9} b=\frac{4+2\sqrt{14}i}{9}$$

Steps for Completing the Square

Add 8b to both sides.

$$-9b^{2}+8b=8$$

Divide both sides by -9.

$$\frac{-9b^{2}+8b}{-9}=\frac{8}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$b^{2}+\frac{8}{-9}b=\frac{8}{-9}$$

Divide 8 by -9.

$$b^{2}-\frac{8}{9}b=\frac{8}{-9}$$

Divide 8 by -9.

$$b^{2}-\frac{8}{9}b=-\frac{8}{9}$$

Divide -\frac{8}{9}, the coefficient of the x term, by 2 to get -\frac{4}{9}. Then add the square of -\frac{4}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-\frac{8}{9}b+\left(-\frac{4}{9}\right)^{2}=-\frac{8}{9}+\left(-\frac{4}{9}\right)^{2}$$

Square -\frac{4}{9} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-\frac{8}{9}b+\frac{16}{81}=-\frac{8}{9}+\frac{16}{81}$$

Add -\frac{8}{9} to \frac{16}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$b^{2}-\frac{8}{9}b+\frac{16}{81}=-\frac{56}{81}$$

Factor b^{2}-\frac{8}{9}b+\frac{16}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{4}{9}\right)^{2}=-\frac{56}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{4}{9}\right)^{2}}=\sqrt{-\frac{56}{81}}$$

Simplify.

$$b-\frac{4}{9}=\frac{2\sqrt{14}i}{9} b-\frac{4}{9}=-\frac{2\sqrt{14}i}{9}$$

Add \frac{4}{9} to both sides of the equation.

$$b=\frac{4+2\sqrt{14}i}{9} b=\frac{-2\sqrt{14}i+4}{9}$$