Algebra Examples

$\frac{3}{y + 3} + \frac{2 y}{y^{2} + 7 y + 12}$

Step 1

Factor $y^{2} + 7 y + 12$ using the AC method.

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Step 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 $7$ .

$3, 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{y + 3}$ by $\frac{y + 4}{y + 4}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $4$ .

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

Step 4.3

Add $3 y$ and $2 y$ .

$\frac{5 y + 12}{(y + 3) (y + 4)}$