Solve 15=8h^2-2h

$$15 = 8 h ^ { 2 } - 2 h$$

Solve for h

$$h = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$h = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$8h^{2}-2h=15$$

Subtract 15 from both sides.

$$8h^{2}-2h-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8h^{2}+ah+bh-15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=8\left(-15\right)=-120$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Calculate the sum for each pair.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

The solution is the pair that gives sum -2.

$$a=-12 b=10$$

Rewrite 8h^{2}-2h-15 as \left(8h^{2}-12h\right)+\left(10h-15\right).

$$\left(8h^{2}-12h\right)+\left(10h-15\right)$$

Factor out 4h in the first and 5 in the second group.

$$4h\left(2h-3\right)+5\left(2h-3\right)$$

Factor out common term 2h-3 by using distributive property.

$$\left(2h-3\right)\left(4h+5\right)$$

To find equation solutions, solve 2h-3=0 and 4h+5=0.

$$h=\frac{3}{2} h=-\frac{5}{4}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$8h^{2}-2h=15$$

Subtract 15 from both sides.

$$8h^{2}-2h-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -2 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 8\left(-15\right)}}{2\times 8}$$

Square -2.

$$h=\frac{-\left(-2\right)±\sqrt{4-4\times 8\left(-15\right)}}{2\times 8}$$

Multiply -4 times 8.

$$h=\frac{-\left(-2\right)±\sqrt{4-32\left(-15\right)}}{2\times 8}$$

Multiply -32 times -15.

$$h=\frac{-\left(-2\right)±\sqrt{4+480}}{2\times 8}$$

Add 4 to 480.

$$h=\frac{-\left(-2\right)±\sqrt{484}}{2\times 8}$$

Take the square root of 484.

$$h=\frac{-\left(-2\right)±22}{2\times 8}$$

The opposite of -2 is 2.

$$h=\frac{2±22}{2\times 8}$$

Multiply 2 times 8.

$$h=\frac{2±22}{16}$$

Now solve the equation h=\frac{2±22}{16} when ± is plus. Add 2 to 22.

$$h=\frac{24}{16}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

$$h=\frac{3}{2}$$

Now solve the equation h=\frac{2±22}{16} when ± is minus. Subtract 22 from 2.

$$h=-\frac{20}{16}$$

Reduce the fraction \frac{-20}{16} to lowest terms by extracting and canceling out 4.

$$h=-\frac{5}{4}$$

The equation is now solved.

$$h=\frac{3}{2} h=-\frac{5}{4}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$8h^{2}-2h=15$$

Divide both sides by 8.

$$\frac{8h^{2}-2h}{8}=\frac{15}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$h^{2}+\left(-\frac{2}{8}\right)h=\frac{15}{8}$$

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

$$h^{2}-\frac{1}{4}h=\frac{15}{8}$$

Divide -\frac{1}{4}, the coefficient of the x term, by 2 to get -\frac{1}{8}. Then add the square of -\frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}-\frac{1}{4}h+\left(-\frac{1}{8}\right)^{2}=\frac{15}{8}+\left(-\frac{1}{8}\right)^{2}$$

Square -\frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$h^{2}-\frac{1}{4}h+\frac{1}{64}=\frac{15}{8}+\frac{1}{64}$$

Add \frac{15}{8} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}-\frac{1}{4}h+\frac{1}{64}=\frac{121}{64}$$

Factor h^{2}-\frac{1}{4}h+\frac{1}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h-\frac{1}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h-\frac{1}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$h-\frac{1}{8}=\frac{11}{8} h-\frac{1}{8}=-\frac{11}{8}$$

Add \frac{1}{8} to both sides of the equation.

$$h=\frac{3}{2} h=-\frac{5}{4}$$