Solve for D ((20y^2-80)/D)÷((4y^2-8y)/(y^3+9y^2))=1

$\frac{20 y^{2} - 80}{D} \div \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1$

Multiply both sides by $\frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$ .

$\frac{20 y^{2} - 80}{D} \div \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify.

Simplify the left side.

Simplify $\frac{20 y^{2} - 80}{D} \div \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$ .

To divide by a fraction, multiply by its reciprocal.

$\frac{20 y^{2} - 80}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify the numerator.

Factor $20$ out of $20 y^{2} - 80$ .

Factor $20$ out of $20 y^{2}$ .

$\frac{20 (y^{2}) - 80}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $20$ out of $- 80$ .

$\frac{20 y^{2} + 20 \cdot - 4}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $20$ out of $20 y^{2} + 20 \cdot - 4$ .

$\frac{20 (y^{2} - 4)}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite $4$ as $2^{2}$ .

$\frac{20 (y^{2} - 2^{2})}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 2$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{3} + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify terms.

Factor $y^{2}$ out of $y^{3} + 9 y^{2}$ .

Factor $y^{2}$ out of $y^{3}$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} y + 9 y^{2}}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $9 y^{2}$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} y + y^{2} \cdot 9}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $y^{2} y + y^{2} \cdot 9$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} (y + 9)}{4 y^{2} - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $4 y^{2} - 8 y$ .

Factor $4 y$ out of $4 y^{2}$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} (y + 9)}{4 y (y) - 8 y} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $- 8 y$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} (y + 9)}{4 y (y) + 4 y (- 2)} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $4 y (y) + 4 y (- 2)$ .

$\frac{20 (y + 2) (y - 2)}{D} \cdot \frac{y^{2} (y + 9)}{4 y (y - 2)} \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Combine.

$\frac{20 (y + 2) (y - 2) (y^{2} (y + 9))}{D (4 y (y - 2))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor of $20$ and $4$ .

Factor $4$ out of $20 (y + 2) (y - 2) (y^{2} (y + 9))$ .

$\frac{4 (5 (y + 2) (y - 2) (y^{2} (y + 9)))}{D (4 y (y - 2))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factors.

Factor $4$ out of $D (4 y (y - 2))$ .

$\frac{4 (5 (y + 2) (y - 2) (y^{2} (y + 9)))}{4 (D (y (y - 2)))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor.

$\frac{4 (5 (y + 2) (y - 2) (y^{2} (y + 9)))}{4 (D (y (y - 2)))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite the expression.

$\frac{5 (y + 2) (y - 2) (y^{2} (y + 9))}{D (y (y - 2))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor of $y - 2$ .

Cancel the common factor.

$\frac{5 (y + 2) (y - 2) (y^{2} (y + 9))}{D (y (y - 2))} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite the expression.

$\frac{5 (y + 2) (y^{2} (y + 9))}{D (y)} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor of $y^{2}$ and $y$ .

Factor $y$ out of $5 (y + 2) (y^{2} (y + 9))$ .

$\frac{y (5 (y + 2) (y (y + 9)))}{D (y)} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factors.

Factor $y$ out of $D (y)$ .

$\frac{y (5 (y + 2) (y (y + 9)))}{y D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor.

$\frac{y (5 (y + 2) (y (y + 9)))}{y D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite the expression.

$\frac{5 (y + 2) (y (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify the numerator.

Rewrite.

$\frac{20 (y + 2) (y - 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4$ out of $20$ .

$\frac{4 (5) (y + 2) (y - 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite.

$\frac{\frac{20}{4} (y + 2) (y - 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Divide $20$ by $4$ .

$\frac{5 (y + 2) (y - 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite.

$\frac{20 (y + 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4$ out of $20$ .

$\frac{4 (5) (y + 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite.

$\frac{\frac{20}{4} (y + 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Divide $20$ by $4$ .

$\frac{5 (y + 2) (y^{2} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y$ out of $y^{2}$ .

$\frac{5 (y + 2) (y \cdot y (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite.

$\frac{5 (y + 2) (y^{1} (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify.

$\frac{5 (y + 2) (y (y + 9))}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Remove unnecessary parentheses.

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify terms.

Factor $4 y$ out of $4 y^{2} - 8 y$ .

Factor $4 y$ out of $4 y^{2}$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y) - 8 y}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $- 8 y$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y) + 4 y (- 2)}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $4 y (y) + 4 y (- 2)$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y - 2)}{y^{3} + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $y^{3} + 9 y^{2}$ .

Factor $y^{2}$ out of $y^{3}$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y - 2)}{y^{2} y + 9 y^{2}} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $9 y^{2}$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y - 2)}{y^{2} y + y^{2} \cdot 9} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $y^{2} y + y^{2} \cdot 9$ .

$\frac{5 (y + 2) y (y + 9)}{D} \cdot \frac{4 y (y - 2)}{y^{2} (y + 9)} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Combine.

$\frac{5 (y + 2) y (y + 9) (4 y (y - 2))}{D (y^{2} (y + 9))} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{5 (y + 2) (y \cdot y) (y + 9) (4 (y - 2))}{D (y^{2} (y + 9))} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Multiply $y$ by $y$ .

$\frac{5 (y + 2) y^{2} (y + 9) (4 (y - 2))}{D (y^{2} (y + 9))} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor of $y^{2}$ .

Cancel the common factor.

$\frac{5 (y + 2) y^{2} (y + 9) (4 (y - 2))}{D (y^{2} (y + 9))} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite the expression.

$\frac{(5 (y + 2) (y + 9)) (4 (y - 2))}{D (y + 9)} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Cancel the common factor of $y + 9$ .

Cancel the common factor.

$\frac{5 (y + 2) (y + 9) (4 (y - 2))}{D (y + 9)} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Rewrite the expression.

$\frac{(5 (y + 2)) (4 (y - 2))}{D} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Multiply $4$ by $5$ .

$\frac{20 (y + 2) (y - 2)}{D} = 1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Simplify the right side.

Simplify $1 \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$ .

Multiply $\frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$ by $1$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y^{2} - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $4 y^{2} - 8 y$ .

Factor $4 y$ out of $4 y^{2}$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y) - 8 y}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $- 8 y$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y) + 4 y (- 2)}{y^{3} + 9 y^{2}}$

Factor $4 y$ out of $4 y (y) + 4 y (- 2)$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y - 2)}{y^{3} + 9 y^{2}}$

Factor $y^{2}$ out of $y^{3} + 9 y^{2}$ .

Factor $y^{2}$ out of $y^{3}$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y - 2)}{y^{2} y + 9 y^{2}}$

Factor $y^{2}$ out of $9 y^{2}$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y - 2)}{y^{2} y + y^{2} \cdot 9}$

Factor $y^{2}$ out of $y^{2} y + y^{2} \cdot 9$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 y (y - 2)}{y^{2} (y + 9)}$

Cancel the common factor of $y$ and $y^{2}$ .

Factor $y$ out of $4 y (y - 2)$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{y (4 (y - 2))}{y^{2} (y + 9)}$

Cancel the common factors.

Factor $y$ out of $y^{2} (y + 9)$ .

$\frac{20 (y + 2) (y - 2)}{D} = \frac{y (4 (y - 2))}{y (y (y + 9))}$

Cancel the common factor.

$\frac{20 (y + 2) (y - 2)}{D} = \frac{y (4 (y - 2))}{y (y (y + 9))}$

Rewrite the expression.

$\frac{20 (y + 2) (y - 2)}{D} = \frac{4 (y - 2)}{y (y + 9)}$

Solve for $D$ .

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$20 (y + 2) (y - 2) (y (y + 9)) = D (4 (y - 2))$

Solve the equation for $D$ .

Rewrite the equation as $D \cdot (4 (y - 2)) = 20 (y + 2) (y - 2) (y (y + 9))$ .

$D \cdot (4 (y - 2)) = 20 (y + 2) (y - 2) (y (y + 9))$

Divide each term in $(D \cdot (4 (y - 2))) = 20 (y + 2) (y - 2) (y (y + 9))$ by $4 (y - 2)$ and simplify.

Divide each term in $(D \cdot (4 (y - 2))) = 20 (y + 2) (y - 2) (y (y + 9))$ by $4 (y - 2)$ .

$\frac{(D \cdot (4 (y - 2)))}{4 (y - 2)} = \frac{20 (y + 2) (y - 2) (y (y + 9))}{4 (y - 2)}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{D \cdot 4 (y - 2)}{4 (y - 2)} = \frac{20 (y + 2) (y - 2) (y (y + 9))}{4 (y - 2)}$

Rewrite the expression.

$\frac{D (y - 2)}{y - 2} = \frac{20 (y + 2) (y - 2) (y (y + 9))}{4 (y - 2)}$

Cancel the common factor of $y - 2$ .

Cancel the common factor.

$\frac{D (y - 2)}{y - 2} = \frac{20 (y + 2) (y - 2) (y (y + 9))}{4 (y - 2)}$

Divide $D$ by $1$ .

$D = \frac{20 (y + 2) (y - 2) (y (y + 9))}{4 (y - 2)}$

Simplify the right side.

Simplify terms.

Cancel the common factor of $20$ and $4$ .

Factor $4$ out of $20 (y + 2) (y - 2) (y (y + 9))$ .

$D = \frac{4 (5 (y + 2) (y - 2) (y (y + 9)))}{4 (y - 2)}$

Cancel the common factors.

Cancel the common factor.

$D = \frac{4 (5 (y + 2) (y - 2) (y (y + 9)))}{4 (y - 2)}$

Rewrite the expression.

$D = \frac{5 (y + 2) (y - 2) (y (y + 9))}{y - 2}$

Cancel the common factor of $y - 2$ .

Cancel the common factor.

$D = \frac{5 (y + 2) (y - 2) (y (y + 9))}{y - 2}$

Divide $5 (y + 2) (y (y + 9))$ by $1$ .

$D = 5 (y + 2) (y (y + 9))$

Apply the distributive property.

$D = (5 y + 5 \cdot 2) (y (y + 9))$

Multiply $5$ by $2$ .

$D = (5 y + 10) (y (y + 9))$

Apply the distributive property.

$D = (5 y + 10) (y \cdot y + y \cdot 9)$

Simplify the expression.

Multiply $y$ by $y$ .

$D = (5 y + 10) (y^{2} + y \cdot 9)$

Move $9$ to the left of $y$ .

$D = (5 y + 10) (y^{2} + 9 \cdot y)$

Expand $(5 y + 10) (y^{2} + 9 y)$ using the FOIL Method.

Apply the distributive property.

$D = 5 y (y^{2} + 9 y) + 10 (y^{2} + 9 y)$

Apply the distributive property.

$D = 5 y \cdot y^{2} + 5 y (9 y) + 10 (y^{2} + 9 y)$

Apply the distributive property.

$D = 5 y \cdot y^{2} + 5 y (9 y) + 10 y^{2} + 10 (9 y)$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$D = 5 (y^{2} y) + 5 y (9 y) + 10 y^{2} + 10 (9 y)$

Multiply $y^{2}$ by $y$ .

Raise $y$ to the power of $1$ .

$D = 5 (y^{2} y^{1}) + 5 y (9 y) + 10 y^{2} + 10 (9 y)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$D = 5 y^{2 + 1} + 5 y (9 y) + 10 y^{2} + 10 (9 y)$

Add $2$ and $1$ .

$D = 5 y^{3} + 5 y (9 y) + 10 y^{2} + 10 (9 y)$

Rewrite using the commutative property of multiplication.

$D = 5 y^{3} + 5 \cdot 9 y \cdot y + 10 y^{2} + 10 (9 y)$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$D = 5 y^{3} + 5 \cdot 9 (y \cdot y) + 10 y^{2} + 10 (9 y)$

Multiply $y$ by $y$ .

$D = 5 y^{3} + 5 \cdot 9 y^{2} + 10 y^{2} + 10 (9 y)$

Multiply $5$ by $9$ .

$D = 5 y^{3} + 45 y^{2} + 10 y^{2} + 10 (9 y)$

Multiply $9$ by $10$ .

$D = 5 y^{3} + 45 y^{2} + 10 y^{2} + 90 y$

Add $45 y^{2}$ and $10 y^{2}$ .

$D = 5 y^{3} + 55 y^{2} + 90 y$

Solve for D ((20y^2-80)/D)÷((4y^2-8y)/(y^3+9y^2))=1

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