Simplify (y^2+8y)/(y^2-9)+(2y+21)/(y^2-9)

$\frac{y^{2} + 8 y}{y^{2} - 9} + \frac{2 y + 21}{y^{2} - 9}$

Step 1

Simplify terms.

Combine the numerators over the common denominator.

$\frac{y^{2} + 8 y + 2 y + 21}{y^{2} - 9}$

Add $8 y$ and $2 y$ .

$\frac{y^{2} + 10 y + 21}{y^{2} - 9}$

Step 2

Factor $y^{2} + 10 y + 21$ using the AC method.

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 $21$ and whose sum is $10$ .

$3, 7$

Write the factored form using these integers.

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

Step 3

Simplify the denominator.

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

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

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 $b = 3$ .

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

Step 4

Cancel the common factor of $y + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (y^2+8y)/(y^2-9)+(2y+21)/(y^2-9)

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