Algebra Examples

$\frac{a + 15}{9 - a^{2}} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{a + 15}{3^{2} - a^{2}} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = a$ .

$\frac{a + 15}{(3 + a) (3 - a)} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Step 2

Reorder terms.

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{a + 5}{a + 3} + \frac{a - 5}{3 - a}$

Step 3

To write $- \frac{a + 5}{a + 3}$ as a fraction with a common denominator, multiply by $\frac{3 - a}{3 - a}$ .

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{a + 5}{a + 3} \cdot \frac{3 - a}{3 - a} + \frac{a - 5}{3 - a}$

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{a + 5}{a + 3}$ by $\frac{3 - a}{3 - a}$ .

$\frac{a + 15}{(a + 3) (3 - a)} - \frac{(a + 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{a + 15 - (a + 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5

Simplify each term.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{a + 15 + (- a - 1 \cdot 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.2

Multiply $- 1$ by $5$ .

$\frac{a + 15 + (- a - 5) (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.3

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

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Step 5.1.3.1

Apply the distributive property.

$\frac{a + 15 - a (3 - a) - 5 (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.3.2

Apply the distributive property.

$\frac{a + 15 - a \cdot 3 - a (- a) - 5 (3 - a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.3.3

Apply the distributive property.

$\frac{a + 15 - a \cdot 3 - a (- a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4

Simplify and combine like terms.

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Step 5.1.4.1

Simplify each term.

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Step 5.1.4.1.1

Multiply $3$ by $- 1$ .

$\frac{a + 15 - 3 a - a (- a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.1.2

Rewrite using the commutative property of multiplication.

$\frac{a + 15 - 3 a - 1 \cdot - 1 a \cdot a - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.1.3

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

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Step 5.1.4.1.3.1

Move $a$ .

$\frac{a + 15 - 3 a - 1 \cdot - 1 (a \cdot a) - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.1.3.2

Multiply $a$ by $a$ .

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

Step 5.1.4.1.4

Multiply $- 1$ by $- 1$ .

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

Step 5.1.4.1.5

Multiply $a^{2}$ by $1$ .

$\frac{a + 15 - 3 a + a^{2} - 5 \cdot 3 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.1.6

Multiply $- 5$ by $3$ .

$\frac{a + 15 - 3 a + a^{2} - 15 - 5 (- a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.1.7

Multiply $- 1$ by $- 5$ .

$\frac{a + 15 - 3 a + a^{2} - 15 + 5 a}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.4.2

Add $- 3 a$ and $5 a$ .

$\frac{a + 15 + 2 a + a^{2} - 15}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.5

Add $a$ and $2 a$ .

$\frac{3 a + 15 + a^{2} - 15}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.6

Subtract $15$ from $15$ .

$\frac{3 a + a^{2} + 0}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.7

Add $3 a + a^{2}$ and $0$ .

$\frac{3 a + a^{2}}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.8

Factor $a$ out of $3 a + a^{2}$ .

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Step 5.1.8.1

Factor $a$ out of $3 a$ .

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

Step 5.1.8.2

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

$\frac{a \cdot 3 + a \cdot a}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.1.8.3

Factor $a$ out of $a \cdot 3 + a \cdot a$ .

$\frac{a (3 + a)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.2

Cancel the common factor of $3 + a$ and $a + 3$ .

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Step 5.2.1

Reorder terms.

$\frac{a (a + 3)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.2.2

Cancel the common factor.

$\frac{a (a + 3)}{(a + 3) (3 - a)} + \frac{a - 5}{3 - a}$

Step 5.2.3

Rewrite the expression.

$\frac{a}{3 - a} + \frac{a - 5}{3 - a}$

Step 6

Simplify terms.

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Step 6.1

Combine the numerators over the common denominator.

$\frac{a + a - 5}{- a + 3}$

Step 6.2

Add $a$ and $a$ .

$\frac{2 a - 5}{- a + 3}$

Step 6.3

Factor $- 1$ out of $- a$ .

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

Step 6.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 6.5

Factor $- 1$ out of $- (a) - 1 (- 3)$ .

$\frac{2 a - 5}{- (a - 3)}$

Step 6.6

Rewrite negatives.

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Step 6.6.1

Rewrite $- (a - 3)$ as $- 1 (a - 3)$ .

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

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{2 a - 5}{a - 3}$