Algebra Examples

$\frac{7}{y + 8} + \frac{35}{y^{2} + 11 y + 24}$

Step 1

Factor $y^{2} + 11 y + 24$ 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 $24$ and whose sum is $11$ .

$3, 8$

Step 1.2

Write the factored form using these integers.

$\frac{7}{y + 8} + \frac{35}{(y + 3) (y + 8)}$

Step 2

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

$\frac{7}{y + 8} \cdot \frac{y + 3}{y + 3} + \frac{35}{(y + 3) (y + 8)}$

Step 3

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

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Step 3.1

Multiply $\frac{7}{y + 8}$ by $\frac{y + 3}{y + 3}$ .

$\frac{7 (y + 3)}{(y + 8) (y + 3)} + \frac{35}{(y + 3) (y + 8)}$

Step 3.2

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

$\frac{7 (y + 3)}{(y + 3) (y + 8)} + \frac{35}{(y + 3) (y + 8)}$

Step 4

Combine the numerators over the common denominator.

$\frac{7 (y + 3) + 35}{(y + 3) (y + 8)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $7$ out of $7 (y + 3) + 35$ .

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Step 5.1.1

Factor $7$ out of $35$ .

$\frac{7 (y + 3) + 7 \cdot 5}{(y + 3) (y + 8)}$

Step 5.1.2

Factor $7$ out of $7 (y + 3) + 7 \cdot 5$ .

$\frac{7 (y + 3 + 5)}{(y + 3) (y + 8)}$

Step 5.2

Add $3$ and $5$ .

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

Step 6

Cancel the common factor of $y + 8$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{7}{y + 3}$