Algebra Examples

$\frac{x + 3}{x^{2} - 25} - \frac{x - 1}{x - 5} + \frac{3}{x + 3}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{x + 3}{x^{2} - 5^{2}} - \frac{x - 1}{x - 5} + \frac{3}{x + 3}$

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 $a = x$ and $b = 5$ .

$\frac{x + 3}{(x + 5) (x - 5)} - \frac{x - 1}{x - 5} + \frac{3}{x + 3}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{x + 3}{(x + 5) (x - 5)}$ by $\frac{x + 3}{x + 3}$ .

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

Step 2.2

Multiply $\frac{x + 3}{(x + 5) (x - 5)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x + 3) (x + 3)}{(x + 5) (x - 5) (x + 3)} - \frac{x - 1}{x - 5} + \frac{3}{x + 3}$

Step 2.3

Multiply $\frac{x - 1}{x - 5}$ by $\frac{(x + 5) (x + 3)}{(x + 5) (x + 3)}$ .

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

Step 2.4

Multiply $\frac{x - 1}{x - 5}$ by $\frac{(x + 5) (x + 3)}{(x + 5) (x + 3)}$ .

$\frac{(x + 3) (x + 3)}{(x + 5) (x - 5) (x + 3)} - \frac{(x - 1) ((x + 5) (x + 3))}{(x - 5) ((x + 5) (x + 3))} + \frac{3}{x + 3}$

Step 2.5

Multiply $\frac{3}{x + 3}$ by $\frac{(x + 5) (x - 5)}{(x + 5) (x - 5)}$ .

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

Step 2.6

Multiply $\frac{3}{x + 3}$ by $\frac{(x + 5) (x - 5)}{(x + 5) (x - 5)}$ .

$\frac{(x + 3) (x + 3)}{(x + 5) (x - 5) (x + 3)} - \frac{(x - 1) ((x + 5) (x + 3))}{(x - 5) ((x + 5) (x + 3))} + \frac{3 ((x + 5) (x - 5))}{(x + 3) ((x + 5) (x - 5))}$

Step 2.7

Reorder the factors of $(x - 5) ((x + 5) (x + 3))$ .

$\frac{(x + 3) (x + 3)}{(x + 5) (x - 5) (x + 3)} - \frac{(x - 1) ((x + 5) (x + 3))}{(x + 3) (x + 5) (x - 5)} + \frac{3 ((x + 5) (x - 5))}{(x + 3) ((x + 5) (x - 5))}$

Step 2.8

Reorder the factors of $(x + 3) ((x + 5) (x - 5))$ .

$\frac{(x + 3) (x + 3)}{(x + 5) (x - 5) (x + 3)} - \frac{(x - 1) ((x + 5) (x + 3))}{(x + 3) (x + 5) (x - 5)} + \frac{3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 3

Combine the numerators over the common denominator.

$\frac{(x + 3) (x + 3) - (x - 1) ((x + 5) (x + 3)) + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4

Simplify each term.

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Step 4.1

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{x (x + 3) + 3 (x + 3) - (x - 1) ((x + 5) (x + 3)) + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

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

Step 4.2.1.2

Move $3$ to the left of $x$ .

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

Step 4.2.1.3

Multiply $3$ by $3$ .

$\frac{x^{2} + 3 x + 3 x + 9 - (x - 1) ((x + 5) (x + 3)) + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.2.2

Add $3 x$ and $3 x$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

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

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Step 4.5.1

Apply the distributive property.

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

Step 4.5.2

Apply the distributive property.

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

Step 4.5.3

Apply the distributive property.

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

Step 4.6

Simplify and combine like terms.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

Multiply $x$ by $x$ .

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

Step 4.6.1.2

Move $3$ to the left of $x$ .

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

Step 4.6.1.3

Multiply $5$ by $3$ .

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

Step 4.6.2

Add $3 x$ and $5 x$ .

$\frac{x^{2} + 6 x + 9 + (- x + 1) (x^{2} + 8 x + 15) + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.7

Expand $(- x + 1) (x^{2} + 8 x + 15)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} + 6 x + 9 - x \cdot x^{2} - x (8 x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8

Simplify each term.

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Step 4.8.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.8.1.1

Move $x^{2}$ .

$\frac{x^{2} + 6 x + 9 - (x^{2} x) - x (8 x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.1.2

Multiply $x^{2}$ by $x$ .

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Step 4.8.1.2.1

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

$\frac{x^{2} + 6 x + 9 - (x^{2} x^{1}) - x (8 x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2} + 6 x + 9 - x^{2 + 1} - x (8 x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.1.3

Add $2$ and $1$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - x (8 x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.2

Rewrite using the commutative property of multiplication.

$\frac{x^{2} + 6 x + 9 - x^{3} - 1 \cdot 8 x \cdot x - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.8.3.1

Move $x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 1 \cdot 8 (x \cdot x) - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.3.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 1 \cdot 8 x^{2} - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.4

Multiply $- 1$ by $8$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 8 x^{2} - x \cdot 15 + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.5

Multiply $15$ by $- 1$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 8 x^{2} - 15 x + 1 x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.6

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

$\frac{x^{2} + 6 x + 9 - x^{3} - 8 x^{2} - 15 x + x^{2} + 1 (8 x) + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.7

Multiply $8 x$ by $1$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 8 x^{2} - 15 x + x^{2} + 8 x + 1 \cdot 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.8.8

Multiply $15$ by $1$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 8 x^{2} - 15 x + x^{2} + 8 x + 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.9

Add $- 8 x^{2}$ and $x^{2}$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 15 x + 8 x + 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.10

Add $- 15 x$ and $8 x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 ((x + 5) (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.11

Expand $(x + 5) (x - 5)$ using the FOIL Method.

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Step 4.11.1

Apply the distributive property.

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x (x - 5) + 5 (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.11.2

Apply the distributive property.

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x \cdot x + x \cdot - 5 + 5 (x - 5))}{(x + 3) (x + 5) (x - 5)}$

Step 4.11.3

Apply the distributive property.

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5)}{(x + 3) (x + 5) (x - 5)}$

Step 4.12

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

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Step 4.12.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x \cdot x - 5 x + 5 x + 5 \cdot - 5)}{(x + 3) (x + 5) (x - 5)}$

Step 4.12.2

Add $- 5 x$ and $5 x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x \cdot x + 0 + 5 \cdot - 5)}{(x + 3) (x + 5) (x - 5)}$

Step 4.12.3

Add $x \cdot x$ and $0$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x \cdot x + 5 \cdot - 5)}{(x + 3) (x + 5) (x - 5)}$

Step 4.13

Simplify each term.

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Step 4.13.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x^{2} + 5 \cdot - 5)}{(x + 3) (x + 5) (x - 5)}$

Step 4.13.2

Multiply $5$ by $- 5$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 (x^{2} - 25)}{(x + 3) (x + 5) (x - 5)}$

Step 4.14

Apply the distributive property.

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 x^{2} + 3 \cdot - 25}{(x + 3) (x + 5) (x - 5)}$

Step 4.15

Multiply $3$ by $- 25$ .

$\frac{x^{2} + 6 x + 9 - x^{3} - 7 x^{2} - 7 x + 15 + 3 x^{2} - 75}{(x + 3) (x + 5) (x - 5)}$

Step 5

Simplify terms.

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Step 5.1

Subtract $7 x^{2}$ from $x^{2}$ .

$\frac{- 6 x^{2} + 6 x + 9 - x^{3} - 7 x + 15 + 3 x^{2} - 75}{(x + 3) (x + 5) (x - 5)}$

Step 5.2

Add $- 6 x^{2}$ and $3 x^{2}$ .

$\frac{- 3 x^{2} + 6 x + 9 - x^{3} - 7 x + 15 - 75}{(x + 3) (x + 5) (x - 5)}$

Step 5.3

Subtract $7 x$ from $6 x$ .

$\frac{- 3 x^{2} - x + 9 - x^{3} + 15 - 75}{(x + 3) (x + 5) (x - 5)}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $9$ and $15$ .

$\frac{- 3 x^{2} - x - x^{3} + 24 - 75}{(x + 3) (x + 5) (x - 5)}$

Step 5.4.2

Subtract $75$ from $24$ .

$\frac{- 3 x^{2} - x - x^{3} - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.4.3

Move $- x$ .

$\frac{- 3 x^{2} - x^{3} - x - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.4.4

Reorder $- 3 x^{2}$ and $- x^{3}$ .

$\frac{- x^{3} - 3 x^{2} - x - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.5

Factor $- 1$ out of $- x^{3}$ .

$\frac{- (x^{3}) - 3 x^{2} - x - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.6

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{- (x^{3}) - (3 x^{2}) - x - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.7

Factor $- 1$ out of $- (x^{3}) - (3 x^{2})$ .

$\frac{- (x^{3} + 3 x^{2}) - x - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.8

Factor $- 1$ out of $- x$ .

$\frac{- (x^{3} + 3 x^{2}) - (x) - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.9

Factor $- 1$ out of $- (x^{3} + 3 x^{2}) - (x)$ .

$\frac{- (x^{3} + 3 x^{2} + x) - 51}{(x + 3) (x + 5) (x - 5)}$

Step 5.10

Rewrite $- 51$ as $- 1 (51)$ .

$\frac{- (x^{3} + 3 x^{2} + x) - 1 (51)}{(x + 3) (x + 5) (x - 5)}$

Step 5.11

Factor $- 1$ out of $- (x^{3} + 3 x^{2} + x) - 1 (51)$ .

$\frac{- (x^{3} + 3 x^{2} + x + 51)}{(x + 3) (x + 5) (x - 5)}$

Step 5.12

Simplify the expression.

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Step 5.12.1

Rewrite $- (x^{3} + 3 x^{2} + x + 51)$ as $- 1 (x^{3} + 3 x^{2} + x + 51)$ .

$\frac{- 1 (x^{3} + 3 x^{2} + x + 51)}{(x + 3) (x + 5) (x - 5)}$

Step 5.12.2

Move the negative in front of the fraction.

$- \frac{x^{3} + 3 x^{2} + x + 51}{(x + 3) (x + 5) (x - 5)}$