Algebra Examples

$\frac{y^{2} - 2 y - 15}{4 y^{2} + 8 y} \cdot \frac{4 y - 20}{y + 3}$

Step 1

Factor $y^{2} - 2 y - 15$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 5) (y + 3)}{4 y^{2} + 8 y} \cdot \frac{4 y - 20}{y + 3}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

$\frac{(y - 5) (y + 3)}{4 y (y) + 8 y} \cdot \frac{4 y - 20}{y + 3}$

Step 2.1.2

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

$\frac{(y - 5) (y + 3)}{4 y (y) + 4 y (2)} \cdot \frac{4 y - 20}{y + 3}$

Step 2.1.3

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

$\frac{(y - 5) (y + 3)}{4 y (y + 2)} \cdot \frac{4 y - 20}{y + 3}$

Step 2.2

Factor $4$ out of $4 y - 20$ .

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Step 2.2.1

Factor $4$ out of $4 y$ .

$\frac{(y - 5) (y + 3)}{4 y (y + 2)} \cdot \frac{4 (y) - 20}{y + 3}$

Step 2.2.2

Factor $4$ out of $- 20$ .

$\frac{(y - 5) (y + 3)}{4 y (y + 2)} \cdot \frac{4 y + 4 \cdot - 5}{y + 3}$

Step 2.2.3

Factor $4$ out of $4 y + 4 \cdot - 5$ .

$\frac{(y - 5) (y + 3)}{4 y (y + 2)} \cdot \frac{4 (y - 5)}{y + 3}$

Step 2.3

Cancel the common factor of $y + 3$ .

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Step 2.3.1

Factor $y + 3$ out of $(y - 5) (y + 3)$ .

$\frac{(y + 3) (y - 5)}{4 y (y + 2)} \cdot \frac{4 (y - 5)}{y + 3}$

Step 2.3.2

Cancel the common factor.

$\frac{(y + 3) (y - 5)}{4 y (y + 2)} \cdot \frac{4 (y - 5)}{y + 3}$

Step 2.3.3

Rewrite the expression.

$\frac{y - 5}{4 y (y + 2)} \cdot (4 (y - 5))$

Step 2.4

Cancel the common factor of $4$ .

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Step 2.4.1

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

$\frac{y - 5}{4 (y (y + 2))} \cdot (4 (y - 5))$

Step 2.4.2

Cancel the common factor.

$\frac{y - 5}{4 (y (y + 2))} \cdot (4 (y - 5))$

Step 2.4.3

Rewrite the expression.

$\frac{y - 5}{y (y + 2)} \cdot (y - 5)$

Step 2.5

Multiply $\frac{y - 5}{y (y + 2)}$ by $y - 5$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $y - 5$ to the power of $1$ .

$\frac{{(y - 5)}^{1} (y - 5)}{y (y + 2)}$

Step 3.2

Raise $y - 5$ to the power of $1$ .

$\frac{{(y - 5)}^{1} {(y - 5)}^{1}}{y (y + 2)}$

Step 3.3

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

$\frac{{(y - 5)}^{1 + 1}}{y (y + 2)}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(y - 5)}^{2}}{y (y + 2)}$