Basic Math Examples

$\frac{y + 7}{y^{2} - y - 12} - \frac{2}{y^{2} - 9}$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} - y - 12$ using the AC method.

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Step 1.1.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 1.1.2

Write the factored form using these integers.

$\frac{y + 7}{(y - 4) (y + 3)} - \frac{2}{y^{2} - 9}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{y + 7}{(y - 4) (y + 3)} - \frac{2}{y^{2} - 3^{2}}$

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 $a = y$ and $b = 3$ .

$\frac{y + 7}{(y - 4) (y + 3)} - \frac{2}{(y + 3) (y - 3)}$

Step 2

To write $\frac{y + 7}{(y - 4) (y + 3)}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{y + 7}{(y - 4) (y + 3)} \cdot \frac{y - 3}{y - 3} - \frac{2}{(y + 3) (y - 3)}$

Step 3

To write $- \frac{2}{(y + 3) (y - 3)}$ as a fraction with a common denominator, multiply by $\frac{y - 4}{y - 4}$ .

$\frac{y + 7}{(y - 4) (y + 3)} \cdot \frac{y - 3}{y - 3} - \frac{2}{(y + 3) (y - 3)} \cdot \frac{y - 4}{y - 4}$

Step 4

Write each expression with a common denominator of $(y - 4) (y + 3) (y - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{y + 7}{(y - 4) (y + 3)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{(y + 7) (y - 3)}{(y - 4) (y + 3) (y - 3)} - \frac{2}{(y + 3) (y - 3)} \cdot \frac{y - 4}{y - 4}$

Step 4.2

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

$\frac{(y + 7) (y - 3)}{(y - 4) (y + 3) (y - 3)} - \frac{2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 4.3

Reorder the factors of $(y - 4) (y + 3) (y - 3)$ .

$\frac{(y + 7) (y - 3)}{(y + 3) (y - 3) (y - 4)} - \frac{2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(y + 7) (y - 3) - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(y + 7) (y - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{y (y - 3) + 7 (y - 3) - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 3 + 7 (y - 3) - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 3 + 7 y + 7 \cdot - 3 - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot - 3 + 7 y + 7 \cdot - 3 - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.2.1.2

Move $- 3$ to the left of $y$ .

$\frac{y^{2} - 3 \cdot y + 7 y + 7 \cdot - 3 - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.2.1.3

Multiply $7$ by $- 3$ .

$\frac{y^{2} - 3 y + 7 y - 21 - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.2.2

Add $- 3 y$ and $7 y$ .

$\frac{y^{2} + 4 y - 21 - 2 (y - 4)}{(y + 3) (y - 3) (y - 4)}$

Step 6.3

Apply the distributive property.

$\frac{y^{2} + 4 y - 21 - 2 y - 2 \cdot - 4}{(y + 3) (y - 3) (y - 4)}$

Step 6.4

Multiply $- 2$ by $- 4$ .

$\frac{y^{2} + 4 y - 21 - 2 y + 8}{(y + 3) (y - 3) (y - 4)}$

Step 6.5

Subtract $2 y$ from $4 y$ .

$\frac{y^{2} + 2 y - 21 + 8}{(y + 3) (y - 3) (y - 4)}$

Step 6.6

Add $- 21$ and $8$ .

$\frac{y^{2} + 2 y - 13}{(y + 3) (y - 3) (y - 4)}$