Solve 8h^2+8=6-9h

$$8 h ^ { 2 } + 8 = 6 - 9 h$$

Solve for h

$$h=\frac{\sqrt{17}-9}{16}\approx -0.304805898$$

$$h=\frac{-\sqrt{17}-9}{16}\approx -0.820194102$$

Steps Using the Quadratic Formula

Subtract 6 from both sides.

$$8h^{2}+8-6=-9h$$

Subtract 6 from 8 to get 2.

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

Add 9h to both sides.

$$8h^{2}+2+9h=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$8h^{2}+9h+2=0$$

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

$$h=\frac{-9±\sqrt{9^{2}-4\times 8\times 2}}{2\times 8}$$

Square 9.

$$h=\frac{-9±\sqrt{81-4\times 8\times 2}}{2\times 8}$$

Multiply -4 times 8.

$$h=\frac{-9±\sqrt{81-32\times 2}}{2\times 8}$$

Multiply -32 times 2.

$$h=\frac{-9±\sqrt{81-64}}{2\times 8}$$

Add 81 to -64.

$$h=\frac{-9±\sqrt{17}}{2\times 8}$$

Multiply 2 times 8.

$$h=\frac{-9±\sqrt{17}}{16}$$

Now solve the equation h=\frac{-9±\sqrt{17}}{16} when ± is plus. Add -9 to \sqrt{17}.

$$h=\frac{\sqrt{17}-9}{16}$$

Now solve the equation h=\frac{-9±\sqrt{17}}{16} when ± is minus. Subtract \sqrt{17} from -9.

$$h=\frac{-\sqrt{17}-9}{16}$$

The equation is now solved.

$$h=\frac{\sqrt{17}-9}{16} h=\frac{-\sqrt{17}-9}{16}$$

Steps for Completing the Square

Add 9h to both sides.

$$8h^{2}+8+9h=6$$

Subtract 8 from both sides.

$$8h^{2}+9h=6-8$$

Subtract 8 from 6 to get -2.

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

Divide both sides by 8.

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

Dividing by 8 undoes the multiplication by 8.

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

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

$$h^{2}+\frac{9}{8}h=-\frac{1}{4}$$

Divide \frac{9}{8}, the coefficient of the x term, by 2 to get \frac{9}{16}. Then add the square of \frac{9}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+\frac{9}{8}h+\left(\frac{9}{16}\right)^{2}=-\frac{1}{4}+\left(\frac{9}{16}\right)^{2}$$

Square \frac{9}{16} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+\frac{9}{8}h+\frac{81}{256}=-\frac{1}{4}+\frac{81}{256}$$

Add -\frac{1}{4} to \frac{81}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}+\frac{9}{8}h+\frac{81}{256}=\frac{17}{256}$$

Factor h^{2}+\frac{9}{8}h+\frac{81}{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}.

$$\left(h+\frac{9}{16}\right)^{2}=\frac{17}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{9}{16}\right)^{2}}=\sqrt{\frac{17}{256}}$$

Simplify.

$$h+\frac{9}{16}=\frac{\sqrt{17}}{16} h+\frac{9}{16}=-\frac{\sqrt{17}}{16}$$

Subtract \frac{9}{16} from both sides of the equation.

$$h=\frac{\sqrt{17}-9}{16} h=\frac{-\sqrt{17}-9}{16}$$