Basic Math Examples

$\frac{y - 1}{y^{2} + 4 y + 3} + \frac{9}{y^{2} + y}$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} + 4 y + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.1.2

Write the factored form using these integers.

$\frac{y - 1}{(y + 1) (y + 3)} + \frac{9}{y^{2} + y}$

Step 1.2

Factor $y$ out of $y^{2} + y$ .

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Step 1.2.1

Factor $y$ out of $y^{2}$ .

$\frac{y - 1}{(y + 1) (y + 3)} + \frac{9}{y \cdot y + y}$

Step 1.2.2

Raise $y$ to the power of $1$ .

$\frac{y - 1}{(y + 1) (y + 3)} + \frac{9}{y \cdot y + y^{1}}$

Step 1.2.3

Factor $y$ out of $y^{1}$ .

$\frac{y - 1}{(y + 1) (y + 3)} + \frac{9}{y \cdot y + y \cdot 1}$

Step 1.2.4

Factor $y$ out of $y \cdot y + y \cdot 1$ .

$\frac{y - 1}{(y + 1) (y + 3)} + \frac{9}{y (y + 1)}$

Step 2

To write $\frac{y - 1}{(y + 1) (y + 3)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{y - 1}{(y + 1) (y + 3)} \cdot \frac{y}{y} + \frac{9}{y (y + 1)}$

Step 3

To write $\frac{9}{y (y + 1)}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{y - 1}{(y + 1) (y + 3)} \cdot \frac{y}{y} + \frac{9}{y (y + 1)} \cdot \frac{y + 3}{y + 3}$

Step 4

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

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Step 4.1

Multiply $\frac{y - 1}{(y + 1) (y + 3)}$ by $\frac{y}{y}$ .

$\frac{(y - 1) y}{(y + 1) (y + 3) y} + \frac{9}{y (y + 1)} \cdot \frac{y + 3}{y + 3}$

Step 4.2

Multiply $\frac{9}{y (y + 1)}$ by $\frac{y + 3}{y + 3}$ .

$\frac{(y - 1) y}{(y + 1) (y + 3) y} + \frac{9 (y + 3)}{y (y + 1) (y + 3)}$

Step 4.3

Reorder the factors of $(y + 1) (y + 3) y$ .

$\frac{(y - 1) y}{y (y + 1) (y + 3)} + \frac{9 (y + 3)}{y (y + 1) (y + 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(y - 1) y + 9 (y + 3)}{y (y + 1) (y + 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{y \cdot y - 1 y + 9 (y + 3)}{y (y + 1) (y + 3)}$

Step 6.2

Multiply $y$ by $y$ .

$\frac{y^{2} - 1 y + 9 (y + 3)}{y (y + 1) (y + 3)}$

Step 6.3

Rewrite $- 1 y$ as $- y$ .

$\frac{y^{2} - y + 9 (y + 3)}{y (y + 1) (y + 3)}$

Step 6.4

Apply the distributive property.

$\frac{y^{2} - y + 9 y + 9 \cdot 3}{y (y + 1) (y + 3)}$

Step 6.5

Multiply $9$ by $3$ .

$\frac{y^{2} - y + 9 y + 27}{y (y + 1) (y + 3)}$

Step 6.6

Add $- y$ and $9 y$ .

$\frac{y^{2} + 8 y + 27}{y (y + 1) (y + 3)}$