Pre-Algebra Examples

\frac{3 y^{2} - 19 y}{y^{2} - 25} + \frac{y^{2} - y}{y^{2} - 25}

$\frac{3 y^{2} - 19 y}{y^{2} - 25} + \frac{y^{2} - y}{y^{2} - 25}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

\frac{3 y^{2} - 19 y + y^{2} - y}{y^{2} - 25}

$\frac{3 y^{2} - 19 y + y^{2} - y}{y^{2} - 25}$

Step 1.2

Add

3 y^{2}

$3 y^{2}$ and

y^{2}

$y^{2}$ .

\frac{4 y^{2} - 19 y - y}{y^{2} - 25}

$\frac{4 y^{2} - 19 y - y}{y^{2} - 25}$

Step 1.3

Subtract

y

$y$ from

- 19 y

$- 19 y$ .

\frac{4 y^{2} - 20 y}{y^{2} - 25}

$\frac{4 y^{2} - 20 y}{y^{2} - 25}$

Step 1.4

Factor

4 y

$4 y$ out of

4 y^{2} - 20 y

$4 y^{2} - 20 y$ .

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Step 1.4.1

Factor

4 y

$4 y$ out of

4 y^{2}

$4 y^{2}$ .

\frac{4 y (y) - 20 y}{y^{2} - 25}

$\frac{4 y (y) - 20 y}{y^{2} - 25}$

Step 1.4.2

Factor

4 y

$4 y$ out of

- 20 y

$- 20 y$ .

\frac{4 y (y) + 4 y (- 5)}{y^{2} - 25}

$\frac{4 y (y) + 4 y (- 5)}{y^{2} - 25}$

Step 1.4.3

Factor

4 y

$4 y$ out of

4 y (y) + 4 y (- 5)

$4 y (y) + 4 y (- 5)$ .

\frac{4 y (y - 5)}{y^{2} - 25}

$\frac{4 y (y - 5)}{y^{2} - 25}$

\frac{4 y (y - 5)}{y^{2} - 25}

$\frac{4 y (y - 5)}{y^{2} - 25}$

\frac{4 y (y - 5)}{y^{2} - 25}

$\frac{4 y (y - 5)}{y^{2} - 25}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\frac{4 y (y - 5)}{y^{2} - 5^{2}}

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = y

$a = y$ and

b = 5

$b = 5$ .

\frac{4 y (y - 5)}{(y + 5) (y - 5)}

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

\frac{4 y (y - 5)}{(y + 5) (y - 5)}

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

Step 3

Cancel the common factor of

y - 5

$y - 5$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{4 y}{y + 5}$