Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(a + 5) (a - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Combine the opposite terms in $a \cdot a + a \cdot - 3 + 3 a + 3 \cdot - 3$ .

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Step 5.2.1

Reorder the factors in the terms $a \cdot - 3$ and $3 a$ .

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

Step 5.2.2

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

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

Step 5.2.3

Add $a \cdot a$ and $0$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $a$ by $a$ .

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

Step 5.3.2

Multiply $3$ by $- 3$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $- 1$ by $- 5$ .

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

Step 5.6

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

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Step 5.6.1

Apply the distributive property.

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

Step 5.6.2

Apply the distributive property.

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

Step 5.6.3

Apply the distributive property.

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

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

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

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Step 5.7.1.1.1

Move $a$ .

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

Step 5.7.1.1.2

Multiply $a$ by $a$ .

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

Step 5.7.1.2

Multiply $5$ by $- 1$ .

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

Step 5.7.1.3

Multiply $5$ by $5$ .

$\frac{a^{2} - 9 - a^{2} - 5 a + 5 a + 25}{(a + 5) (a - 3)}$

Step 5.7.2

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

$\frac{a^{2} - 9 - a^{2} + 0 + 25}{(a + 5) (a - 3)}$

Step 5.7.3

Add $- a^{2}$ and $0$ .

$\frac{a^{2} - 9 - a^{2} + 25}{(a + 5) (a - 3)}$

Step 5.8

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

$\frac{0 - 9 + 25}{(a + 5) (a - 3)}$

Step 5.9

Subtract $9$ from $0$ .

$\frac{- 9 + 25}{(a + 5) (a - 3)}$

Step 5.10

Add $- 9$ and $25$ .

$\frac{16}{(a + 5) (a - 3)}$