Basic Math Examples

$\frac{y^{2} - 25}{y^{2} + 5 y} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{y^{2} - 5^{2}}{y^{2} + 5 y} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 1.2

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 $b = 5$ .

$\frac{(y + 5) (y - 5)}{y^{2} + 5 y} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 2

Simplify with factoring out.

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Step 2.1

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

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Step 2.1.1

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

$\frac{(y + 5) (y - 5)}{y \cdot y + 5 y} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 2.1.2

Factor $y$ out of $5 y$ .

$\frac{(y + 5) (y - 5)}{y \cdot y + y \cdot 5} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 2.1.3

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

$\frac{(y + 5) (y - 5)}{y (y + 5)} \cdot \frac{4 y - 12}{y z - 2 y - 5 z + 10}$

Step 2.2

Factor $4$ out of $4 y - 12$ .

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Step 2.2.1

Factor $4$ out of $4 y$ .

$\frac{(y + 5) (y - 5)}{y (y + 5)} \cdot \frac{4 (y) - 12}{y z - 2 y - 5 z + 10}$

Step 2.2.2

Factor $4$ out of $- 12$ .

$\frac{(y + 5) (y - 5)}{y (y + 5)} \cdot \frac{4 y + 4 \cdot - 3}{y z - 2 y - 5 z + 10}$

Step 2.2.3

Factor $4$ out of $4 y + 4 \cdot - 3$ .

$\frac{(y + 5) (y - 5)}{y (y + 5)} \cdot \frac{4 (y - 3)}{y z - 2 y - 5 z + 10}$

Step 3

Simplify the denominator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$\frac{(y + 5) (y - 5)}{y (y + 5)} \cdot \frac{4 (y - 3)}{(y z - 2 y) - 5 z + 10}$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $z - 2$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $y - 5$ .

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Step 4.1.1

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

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

Step 4.1.2

Factor $y - 5$ out of $(z - 2) (y - 5)$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Multiply $\frac{y + 5}{y (y + 5)}$ by $\frac{4 (y - 3)}{z - 2}$ .

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

Step 4.3

Cancel the common factor of $y + 5$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{4 (y - 3)}{y (z - 2)}$