Basic Math Examples

\frac{2 y^{2} - 9 y - 13}{y^{2} + 4 y + 3} - \frac{y}{y + 1} + \frac{12}{y + 3}

$\frac{2 y^{2} - 9 y - 13}{y^{2} + 4 y + 3} - \frac{y}{y + 1} + \frac{12}{y + 3}$

Step 1

Factor

y^{2} + 4 y + 3

$y^{2} + 4 y + 3$ using the AC method.

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

3

$3$ and whose sum is

4

$4$ .

1, 3

$1, 3$

Step 1.2

Write the factored form using these integers.

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y}{y + 1} + \frac{12}{y + 3}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y}{y + 1} + \frac{12}{y + 3}$

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y}{y + 1} + \frac{12}{y + 3}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y}{y + 1} + \frac{12}{y + 3}$

Step 2

Find the common denominator.

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Step 2.1

Multiply

\frac{y}{y + 1}

$\frac{y}{y + 1}$ by

\frac{y + 3}{y + 3}

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

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - (\frac{y}{y + 1} \cdot \frac{y + 3}{y + 3}) + \frac{12}{y + 3}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - (\frac{y}{y + 1} \cdot \frac{y + 3}{y + 3}) + \frac{12}{y + 3}$

Step 2.2

Multiply

\frac{y}{y + 1}

$\frac{y}{y + 1}$ by

\frac{y + 3}{y + 3}

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

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12}{y + 3}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12}{y + 3}$

Step 2.3

Multiply

\frac{12}{y + 3}

$\frac{12}{y + 3}$ by

\frac{y + 1}{y + 1}

$\frac{y + 1}{y + 1}$ .

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12}{y + 3} \cdot \frac{y + 1}{y + 1}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12}{y + 3} \cdot \frac{y + 1}{y + 1}$

Step 2.4

Multiply

\frac{12}{y + 3}

$\frac{12}{y + 3}$ by

\frac{y + 1}{y + 1}

$\frac{y + 1}{y + 1}$ .

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 3) (y + 1)}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 3) (y + 1)}$

Step 2.5

Reorder the factors of

(y + 3) (y + 1)

$(y + 3) (y + 1)$ .

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 1) (y + 3)}$

\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13}{(y + 1) (y + 3)} - \frac{y (y + 3)}{(y + 1) (y + 3)} + \frac{12 (y + 1)}{(y + 1) (y + 3)}$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

\frac{2 y^{2} - 9 y - 13 - y (y + 3) + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y (y + 3) + 12 (y + 1)}{(y + 1) (y + 3)}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

\frac{2 y^{2} - 9 y - 13 - y \cdot y - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y \cdot y - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}$

Step 3.2.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

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Step 3.2.2.1

Move

y

$y$ .

\frac{2 y^{2} - 9 y - 13 - (y \cdot y) - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - (y \cdot y) - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}$

Step 3.2.2.2

Multiply

y

$y$ by

y

$y$ .

\frac{2 y^{2} - 9 y - 13 - y^{2} - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}$

\frac{2 y^{2} - 9 y - 13 - y^{2} - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - y \cdot 3 + 12 (y + 1)}{(y + 1) (y + 3)}$

Step 3.2.3

Multiply

3

$3$ by

- 1

$- 1$ .

\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 (y + 1)}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 (y + 1)}{(y + 1) (y + 3)}$

Step 3.2.4

Apply the distributive property.

\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12 \cdot 1}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12 \cdot 1}{(y + 1) (y + 3)}$

Step 3.2.5

Multiply

12

$12$ by

1

$1$ .

\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12}{(y + 1) (y + 3)}$

\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12}{(y + 1) (y + 3)}

$\frac{2 y^{2} - 9 y - 13 - y^{2} - 3 y + 12 y + 12}{(y + 1) (y + 3)}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Subtract

y^{2}

$y^{2}$ from

2 y^{2}

$2 y^{2}$ .

\frac{y^{2} - 9 y - 13 - 3 y + 12 y + 12}{(y + 1) (y + 3)}

$\frac{y^{2} - 9 y - 13 - 3 y + 12 y + 12}{(y + 1) (y + 3)}$

Step 3.3.2

Subtract

3 y

$3 y$ from

- 9 y

$- 9 y$ .

\frac{y^{2} - 12 y - 13 + 12 y + 12}{(y + 1) (y + 3)}

$\frac{y^{2} - 12 y - 13 + 12 y + 12}{(y + 1) (y + 3)}$

Step 3.3.3

Combine the opposite terms in

y^{2} - 12 y - 13 + 12 y + 12

$y^{2} - 12 y - 13 + 12 y + 12$ .

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Step 3.3.3.1

Add

- 12 y

$- 12 y$ and

12 y

$12 y$ .

\frac{y^{2} + 0 - 13 + 12}{(y + 1) (y + 3)}

$\frac{y^{2} + 0 - 13 + 12}{(y + 1) (y + 3)}$

Step 3.3.3.2

Add

y^{2}

$y^{2}$ and

0

$0$ .

\frac{y^{2} - 13 + 12}{(y + 1) (y + 3)}

$\frac{y^{2} - 13 + 12}{(y + 1) (y + 3)}$

\frac{y^{2} - 13 + 12}{(y + 1) (y + 3)}

$\frac{y^{2} - 13 + 12}{(y + 1) (y + 3)}$

Step 3.3.4

Add

- 13

$- 13$ and

12

$12$ .

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

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

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

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

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

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = y

$a = y$ and

b = 1

$b = 1$ .

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

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

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

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

Step 5

Cancel the common factor of

y + 1

$y + 1$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{y - 1}{y + 3}$