Algebra Examples

$\frac{2 a + 5}{a^{2} + 5 a + 4} + \frac{9}{(a + 4) (a^{2} + 4 a + 3)} - \frac{a b}{a^{2} b + 7 a b + 12 b}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} + 5 a + 4$ 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 $4$ and whose sum is $5$ .

$1, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a^{2} + 4 a + 3)} - \frac{a b}{a^{2} b + 7 a b + 12 b}$

Step 1.2

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

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Step 1.2.1

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 1.2.2

Write the factored form using these integers.

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) ((a + 1) (a + 3))} - \frac{a b}{a^{2} b + 7 a b + 12 b}$

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{a^{2} b + 7 a b + 12 b}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Factor $b$ out of $a^{2} b + 7 a b + 12 b$ .

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Step 1.3.1.1

Factor $b$ out of $a^{2} b$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b a^{2} + 7 a b + 12 b}$

Step 1.3.1.2

Factor $b$ out of $7 a b$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b a^{2} + b (7 a) + 12 b}$

Step 1.3.1.3

Factor $b$ out of $12 b$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b a^{2} + b (7 a) + b \cdot 12}$

Step 1.3.1.4

Factor $b$ out of $b a^{2} + b (7 a)$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b (a^{2} + 7 a) + b \cdot 12}$

Step 1.3.1.5

Factor $b$ out of $b (a^{2} + 7 a) + b \cdot 12$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b (a^{2} + 7 a + 12)}$

Step 1.3.2

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

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Step 1.3.2.1

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 1.3.2.2

Write the factored form using these integers.

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b ((a + 3) (a + 4))}$

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b (a + 3) (a + 4)}$

Step 1.4

Cancel the common factor of $b$ .

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Step 1.4.1

Cancel the common factor.

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a b}{b (a + 3) (a + 4)}$

Step 1.4.2

Rewrite the expression.

$\frac{2 a + 5}{(a + 1) (a + 4)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a}{(a + 3) (a + 4)}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{2 a + 5}{(a + 1) (a + 4)}$ by $\frac{a + 3}{a + 3}$ .

$\frac{2 a + 5}{(a + 1) (a + 4)} \cdot \frac{a + 3}{a + 3} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a}{(a + 3) (a + 4)}$

Step 2.2

Multiply $\frac{2 a + 5}{(a + 1) (a + 4)}$ by $\frac{a + 3}{a + 3}$ .

$\frac{(2 a + 5) (a + 3)}{(a + 1) (a + 4) (a + 3)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a}{(a + 3) (a + 4)}$

Step 2.3

Multiply $\frac{a}{(a + 3) (a + 4)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{(2 a + 5) (a + 3)}{(a + 1) (a + 4) (a + 3)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - (\frac{a}{(a + 3) (a + 4)} \cdot \frac{a + 1}{a + 1})$

Step 2.4

Multiply $\frac{a}{(a + 3) (a + 4)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{(2 a + 5) (a + 3)}{(a + 1) (a + 4) (a + 3)} + \frac{9}{(a + 4) (a + 1) (a + 3)} - \frac{a (a + 1)}{(a + 3) (a + 4) (a + 1)}$

Step 2.5

Reorder the factors of $(a + 4) (a + 1) (a + 3)$ .

$\frac{(2 a + 5) (a + 3)}{(a + 1) (a + 4) (a + 3)} + \frac{9}{(a + 1) (a + 3) (a + 4)} - \frac{a (a + 1)}{(a + 3) (a + 4) (a + 1)}$

Step 2.6

Reorder the factors of $(a + 3) (a + 4) (a + 1)$ .

$\frac{(2 a + 5) (a + 3)}{(a + 1) (a + 4) (a + 3)} + \frac{9}{(a + 1) (a + 3) (a + 4)} - \frac{a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{(2 a + 5) (a + 3) + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2

Simplify each term.

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Step 3.2.1

Expand $(2 a + 5) (a + 3)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

$\frac{2 a (a + 3) + 5 (a + 3) + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.1.2

Apply the distributive property.

$\frac{2 a \cdot a + 2 a \cdot 3 + 5 (a + 3) + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{2 a \cdot a + 2 a \cdot 3 + 5 a + 5 \cdot 3 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 3.2.2.1.1.1

Move $a$ .

$\frac{2 (a \cdot a) + 2 a \cdot 3 + 5 a + 5 \cdot 3 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.2.1.1.2

Multiply $a$ by $a$ .

$\frac{2 a^{2} + 2 a \cdot 3 + 5 a + 5 \cdot 3 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.2.1.2

Multiply $3$ by $2$ .

$\frac{2 a^{2} + 6 a + 5 a + 5 \cdot 3 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.2.1.3

Multiply $5$ by $3$ .

$\frac{2 a^{2} + 6 a + 5 a + 15 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.2.2

Add $6 a$ and $5 a$ .

$\frac{2 a^{2} + 11 a + 15 + 9 - a (a + 1)}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.3

Apply the distributive property.

$\frac{2 a^{2} + 11 a + 15 + 9 - a \cdot a - a \cdot 1}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.4

Multiply $a$ by $a$ by adding the exponents.

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Step 3.2.4.1

Move $a$ .

$\frac{2 a^{2} + 11 a + 15 + 9 - (a \cdot a) - a \cdot 1}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.4.2

Multiply $a$ by $a$ .

$\frac{2 a^{2} + 11 a + 15 + 9 - a^{2} - a \cdot 1}{(a + 1) (a + 3) (a + 4)}$

Step 3.2.5

Multiply $- 1$ by $1$ .

$\frac{2 a^{2} + 11 a + 15 + 9 - a^{2} - a}{(a + 1) (a + 3) (a + 4)}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Subtract $a^{2}$ from $2 a^{2}$ .

$\frac{a^{2} + 11 a + 15 + 9 - a}{(a + 1) (a + 3) (a + 4)}$

Step 3.3.2

Subtract $a$ from $11 a$ .

$\frac{a^{2} + 10 a + 15 + 9}{(a + 1) (a + 3) (a + 4)}$

Step 3.3.3

Add $15$ and $9$ .

$\frac{a^{2} + 10 a + 24}{(a + 1) (a + 3) (a + 4)}$

Step 4

Factor $a^{2} + 10 a + 24$ using the AC method.

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Step 4.1

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 $10$ .

$4, 6$

Step 4.2

Write the factored form using these integers.

$\frac{(a + 4) (a + 6)}{(a + 1) (a + 3) (a + 4)}$

Step 5

Cancel the common factor of $a + 4$ .

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Step 5.1

Cancel the common factor.

$\frac{(a + 4) (a + 6)}{(a + 1) (a + 3) (a + 4)}$

Step 5.2

Rewrite the expression.

$\frac{a + 6}{(a + 1) (a + 3)}$