Algebra Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 5$ .

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

Step 1.2

Factor $a$ out of $a^{2} + 5 a$ .

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Factor $a$ out of $5 a$ .

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

Step 1.2.3

Factor $a$ out of $a \cdot a + a \cdot 5$ .

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

Step 1.3

Cancel the common factor of $a + 5$ .

Tap for more steps...

Step 1.3.1

Factor $a + 5$ out of $a (a + 5)$ .

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Multiply $\frac{a - 5}{a + 3}$ by $\frac{1}{a}$ .

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

Step 1.5

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

Tap for more steps...

Step 1.5.1

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

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

Step 1.5.2

Factor $a$ out of $- 3 a$ .

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

Step 1.5.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(a + 3) a (a - 3)$ .

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

Step 4.4

Reorder the factors of $a (a - 3) (a + 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $a$ by $a$ .

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

Step 6.2.1.2

Move $- 3$ to the left of $a$ .

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

Step 6.2.1.3

Multiply $- 5$ by $- 3$ .

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

Step 6.2.2

Subtract $5 a$ from $- 3 a$ .

$\frac{a^{2} - 8 a + 15 - (a + 5) (a + 3)}{a (a + 3) (a - 3)}$

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- 1$ by $5$ .

$\frac{a^{2} - 8 a + 15 + (- a - 5) (a + 3)}{a (a + 3) (a - 3)}$

Step 6.5

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

Tap for more steps...

Step 6.5.1

Apply the distributive property.

$\frac{a^{2} - 8 a + 15 - a (a + 3) - 5 (a + 3)}{a (a + 3) (a - 3)}$

Step 6.5.2

Apply the distributive property.

$\frac{a^{2} - 8 a + 15 - a \cdot a - a \cdot 3 - 5 (a + 3)}{a (a + 3) (a - 3)}$

Step 6.5.3

Apply the distributive property.

$\frac{a^{2} - 8 a + 15 - a \cdot a - a \cdot 3 - 5 a - 5 \cdot 3}{a (a + 3) (a - 3)}$

Step 6.6

Simplify and combine like terms.

Tap for more steps...

Step 6.6.1

Simplify each term.

Tap for more steps...

Step 6.6.1.1

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

Tap for more steps...

Step 6.6.1.1.1

Move $a$ .

$\frac{a^{2} - 8 a + 15 - (a \cdot a) - a \cdot 3 - 5 a - 5 \cdot 3}{a (a + 3) (a - 3)}$

Step 6.6.1.1.2

Multiply $a$ by $a$ .

$\frac{a^{2} - 8 a + 15 - a^{2} - a \cdot 3 - 5 a - 5 \cdot 3}{a (a + 3) (a - 3)}$

Step 6.6.1.2

Multiply $3$ by $- 1$ .

$\frac{a^{2} - 8 a + 15 - a^{2} - 3 a - 5 a - 5 \cdot 3}{a (a + 3) (a - 3)}$

Step 6.6.1.3

Multiply $- 5$ by $3$ .

$\frac{a^{2} - 8 a + 15 - a^{2} - 3 a - 5 a - 15}{a (a + 3) (a - 3)}$

Step 6.6.2

Subtract $5 a$ from $- 3 a$ .

$\frac{a^{2} - 8 a + 15 - a^{2} - 8 a - 15}{a (a + 3) (a - 3)}$

Step 6.7

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

$\frac{- 8 a + 15 + 0 - 8 a - 15}{a (a + 3) (a - 3)}$

Step 6.8

Add $- 8 a$ and $0$ .

$\frac{- 8 a + 15 - 8 a - 15}{a (a + 3) (a - 3)}$

Step 6.9

Subtract $8 a$ from $- 8 a$ .

$\frac{- 16 a + 15 - 15}{a (a + 3) (a - 3)}$

Step 6.10

Subtract $15$ from $15$ .

$\frac{- 16 a + 0}{a (a + 3) (a - 3)}$

Step 6.11

Add $- 16 a$ and $0$ .

$\frac{- 16 a}{a (a + 3) (a - 3)}$

Step 7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 7.1

Cancel the common factor of $a$ .

Tap for more steps...

Step 7.1.1

Cancel the common factor.

$\frac{- 16 a}{a (a + 3) (a - 3)}$

Step 7.1.2

Rewrite the expression.

$\frac{- 16}{(a + 3) (a - 3)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{16}{(a + 3) (a - 3)}$