Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

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

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 $b = 1$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Factor $a$ out of $a^{1}$ .

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

Step 1.2.4

Factor $a$ out of $a \cdot a + a \cdot 1$ .

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

Step 2

To write $\frac{a + 3}{(a + 1) (a - 1)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 3

To write $- \frac{1}{a (a + 1)}$ as a fraction with a common denominator, multiply by $\frac{a - 1}{a - 1}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(a + 1) (a - 1) a$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $a$ by $a$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- 1$ by $- 1$ .

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

Step 6.5

Subtract $a$ from $3 a$ .

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

Step 6.6

Factor using the perfect square rule.

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Step 6.6.1

Rewrite $1$ as $1^{2}$ .

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

Step 6.6.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Step 6.6.3

Rewrite the polynomial.

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

Step 6.6.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

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

Step 7

Cancel the common factor of ${(a + 1)}^{2}$ and $a + 1$ .

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Step 7.1

Factor $a + 1$ out of ${(a + 1)}^{2}$ .

$\frac{(a + 1) (a + 1)}{a (a + 1) (a - 1)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$\frac{(a + 1) (a + 1)}{(a + 1) (a (a - 1))}$

Step 7.2.2

Cancel the common factor.

$\frac{(a + 1) (a + 1)}{(a + 1) (a (a - 1))}$

Step 7.2.3

Rewrite the expression.

$\frac{a + 1}{a (a - 1)}$