Algebra Examples

$\frac{3 y + 2}{y^{2} + 5 y - 24} + \frac{7}{y^{2} + 4 y - 32}$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} + 5 y - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 1.1.2

Write the factored form using these integers.

$\frac{3 y + 2}{(y - 3) (y + 8)} + \frac{7}{y^{2} + 4 y - 32}$

Step 1.2

Factor $y^{2} + 4 y - 32$ 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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{3 y + 2}{(y - 3) (y + 8)} + \frac{7}{(y - 4) (y + 8)}$

Step 2

To write $\frac{3 y + 2}{(y - 3) (y + 8)}$ as a fraction with a common denominator, multiply by $\frac{y - 4}{y - 4}$ .

$\frac{3 y + 2}{(y - 3) (y + 8)} \cdot \frac{y - 4}{y - 4} + \frac{7}{(y - 4) (y + 8)}$

Step 3

To write $\frac{7}{(y - 4) (y + 8)}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{3 y + 2}{(y - 3) (y + 8)} \cdot \frac{y - 4}{y - 4} + \frac{7}{(y - 4) (y + 8)} \cdot \frac{y - 3}{y - 3}$

Step 4

Write each expression with a common denominator of $(y - 3) (y + 8) (y - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 y + 2}{(y - 3) (y + 8)}$ by $\frac{y - 4}{y - 4}$ .

$\frac{(3 y + 2) (y - 4)}{(y - 3) (y + 8) (y - 4)} + \frac{7}{(y - 4) (y + 8)} \cdot \frac{y - 3}{y - 3}$

Step 4.2

Multiply $\frac{7}{(y - 4) (y + 8)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{(3 y + 2) (y - 4)}{(y - 3) (y + 8) (y - 4)} + \frac{7 (y - 3)}{(y - 4) (y + 8) (y - 3)}$

Step 4.3

Reorder the factors of $(y - 3) (y + 8) (y - 4)$ .

$\frac{(3 y + 2) (y - 4)}{(y + 8) (y - 3) (y - 4)} + \frac{7 (y - 3)}{(y - 4) (y + 8) (y - 3)}$

Step 4.4

Reorder the factors of $(y - 4) (y + 8) (y - 3)$ .

$\frac{(3 y + 2) (y - 4)}{(y + 8) (y - 3) (y - 4)} + \frac{7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(3 y + 2) (y - 4) + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(3 y + 2) (y - 4)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{3 y (y - 4) + 2 (y - 4) + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.1.2

Apply the distributive property.

$\frac{3 y \cdot y + 3 y \cdot - 4 + 2 (y - 4) + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.1.3

Apply the distributive property.

$\frac{3 y \cdot y + 3 y \cdot - 4 + 2 y + 2 \cdot - 4 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

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 $y$ by adding the exponents.

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Step 6.2.1.1.1

Move $y$ .

$\frac{3 (y \cdot y) + 3 y \cdot - 4 + 2 y + 2 \cdot - 4 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.2.1.1.2

Multiply $y$ by $y$ .

$\frac{3 y^{2} + 3 y \cdot - 4 + 2 y + 2 \cdot - 4 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.2.1.2

Multiply $- 4$ by $3$ .

$\frac{3 y^{2} - 12 y + 2 y + 2 \cdot - 4 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.2.1.3

Multiply $2$ by $- 4$ .

$\frac{3 y^{2} - 12 y + 2 y - 8 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.2.2

Add $- 12 y$ and $2 y$ .

$\frac{3 y^{2} - 10 y - 8 + 7 (y - 3)}{(y + 8) (y - 3) (y - 4)}$

Step 6.3

Apply the distributive property.

$\frac{3 y^{2} - 10 y - 8 + 7 y + 7 \cdot - 3}{(y + 8) (y - 3) (y - 4)}$

Step 6.4

Multiply $7$ by $- 3$ .

$\frac{3 y^{2} - 10 y - 8 + 7 y - 21}{(y + 8) (y - 3) (y - 4)}$

Step 6.5

Add $- 10 y$ and $7 y$ .

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

Step 6.6

Subtract $21$ from $- 8$ .

$\frac{3 y^{2} - 3 y - 29}{(y + 8) (y - 3) (y - 4)}$