Trigonometry Examples

$\frac{3 y + 15}{25 - y^{2}} + \frac{2}{y - 5}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 y + 15$ .

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Step 1.1.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) + 15}{25 - y^{2}} + \frac{2}{y - 5}$

Step 1.1.2

Factor $3$ out of $15$ .

$\frac{3 y + 3 \cdot 5}{25 - y^{2}} + \frac{2}{y - 5}$

Step 1.1.3

Factor $3$ out of $3 y + 3 \cdot 5$ .

$\frac{3 (y + 5)}{25 - y^{2}} + \frac{2}{y - 5}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = y$ .

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

Step 1.3

Cancel the common factor of $y + 5$ and $5 + y$ .

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Step 1.3.1

Reorder terms.

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

$\frac{3}{5 - y} + \frac{2}{y - 5}$

Step 2

Simplify terms.

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Step 2.1

Factor $- 1$ out of $y$ .

$\frac{3}{5 - y} + \frac{2}{- 1 (- y) - 5}$

Step 2.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{3}{5 - y} + \frac{2}{- 1 (- y) - 1 (5)}$

Step 2.3

Factor $- 1$ out of $- 1 (- y) - 1 (5)$ .

$\frac{3}{5 - y} + \frac{2}{- 1 (- y + 5)}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{2}{- 1 (- y + 5)}$ to the numerator.

$\frac{3}{5 - y} + \frac{- 1 \cdot 2}{- y + 5}$

Step 2.4.2

Reorder terms.

$\frac{3}{- y + 5} + \frac{- 1 \cdot 2}{- y + 5}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 - 1 \cdot 2}{- y + 5}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $- 1$ by $2$ .

$\frac{3 - 2}{- y + 5}$

Step 3.2

Subtract $2$ from $3$ .

$\frac{1}{- y + 5}$

Step 4

Factor $- 1$ out of $- y$ .

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

Step 5

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{1}{- (y) - 1 (- 5)}$

Step 6

Factor $- 1$ out of $- (y) - 1 (- 5)$ .

$\frac{1}{- (y - 5)}$

Step 7

Rewrite negatives.

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Step 7.1

Rewrite $- (y - 5)$ as $- 1 (y - 5)$ .

$\frac{1}{- 1 (y - 5)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{1}{y - 5}$