Basic Math Examples

$\frac{6}{y^{2} + 16 y - 80} - \frac{4}{y^{2} + 24 y + 80}$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} + 16 y - 80$ 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 $- 80$ and whose sum is $16$ .

$- 4, 20$

Step 1.1.2

Write the factored form using these integers.

$\frac{6}{(y - 4) (y + 20)} - \frac{4}{y^{2} + 24 y + 80}$

Step 1.2

Factor $y^{2} + 24 y + 80$ using the AC method.

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Step 1.2.1

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 $80$ and whose sum is $24$ .

$4, 20$

Step 1.2.2

Write the factored form using these integers.

$\frac{6}{(y - 4) (y + 20)} - \frac{4}{(y + 4) (y + 20)}$

Step 2

To write $\frac{6}{(y - 4) (y + 20)}$ as a fraction with a common denominator, multiply by $\frac{y + 4}{y + 4}$ .

$\frac{6}{(y - 4) (y + 20)} \cdot \frac{y + 4}{y + 4} - \frac{4}{(y + 4) (y + 20)}$

Step 3

To write $- \frac{4}{(y + 4) (y + 20)}$ as a fraction with a common denominator, multiply by $\frac{y - 4}{y - 4}$ .

$\frac{6}{(y - 4) (y + 20)} \cdot \frac{y + 4}{y + 4} - \frac{4}{(y + 4) (y + 20)} \cdot \frac{y - 4}{y - 4}$

Step 4

Write each expression with a common denominator of $(y - 4) (y + 20) (y + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6}{(y - 4) (y + 20)}$ by $\frac{y + 4}{y + 4}$ .

$\frac{6 (y + 4)}{(y - 4) (y + 20) (y + 4)} - \frac{4}{(y + 4) (y + 20)} \cdot \frac{y - 4}{y - 4}$

Step 4.2

Multiply $\frac{4}{(y + 4) (y + 20)}$ by $\frac{y - 4}{y - 4}$ .

$\frac{6 (y + 4)}{(y - 4) (y + 20) (y + 4)} - \frac{4 (y - 4)}{(y + 4) (y + 20) (y - 4)}$

Step 4.3

Reorder the factors of $(y - 4) (y + 20) (y + 4)$ .

$\frac{6 (y + 4)}{(y + 20) (y + 4) (y - 4)} - \frac{4 (y - 4)}{(y + 4) (y + 20) (y - 4)}$

Step 4.4

Reorder the factors of $(y + 4) (y + 20) (y - 4)$ .

$\frac{6 (y + 4)}{(y + 20) (y + 4) (y - 4)} - \frac{4 (y - 4)}{(y + 20) (y + 4) (y - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 (y + 4) - 4 (y - 4)}{(y + 20) (y + 4) (y - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $6 (y + 4) - 4 (y - 4)$ .

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Step 6.1.1

Factor $2$ out of $6 (y + 4)$ .

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

Step 6.1.2

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

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

Step 6.1.3

Factor $2$ out of $2 (3 (y + 4)) + 2 (- 2 (y - 4))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $3$ by $4$ .

$\frac{2 (3 y + 12 - 2 (y - 4))}{(y + 20) (y + 4) (y - 4)}$

Step 6.4

Apply the distributive property.

$\frac{2 (3 y + 12 - 2 y - 2 \cdot - 4)}{(y + 20) (y + 4) (y - 4)}$

Step 6.5

Multiply $- 2$ by $- 4$ .

$\frac{2 (3 y + 12 - 2 y + 8)}{(y + 20) (y + 4) (y - 4)}$

Step 6.6

Subtract $2 y$ from $3 y$ .

$\frac{2 (y + 12 + 8)}{(y + 20) (y + 4) (y - 4)}$

Step 6.7

Add $12$ and $8$ .

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

Step 7

Cancel the common factor of $y + 20$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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