Basic Math Examples

$\frac{1}{a^{2} - 1} + \frac{a - 1}{a^{2} + 3 a - 4}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

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

$\frac{1}{a^{2} - 1^{2}} + \frac{a - 1}{a^{2} + 3 a - 4}$

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 $b = 1$ .

$\frac{1}{(a + 1) (a - 1)} + \frac{a - 1}{a^{2} + 3 a - 4}$

Step 1.2

Factor $a^{2} + 3 a - 4$ using the AC method.

Tap for more steps...

Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{1}{(a + 1) (a - 1)} + \frac{a - 1}{(a - 1) (a + 4)}$

Step 1.3

Cancel the common factor of $a - 1$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{1}{(a + 1) (a - 1)} + \frac{a - 1}{(a - 1) (a + 4)}$

Step 1.3.2

Rewrite the expression.

$\frac{1}{(a + 1) (a - 1)} + \frac{1}{a + 4}$

Step 2

To write $\frac{1}{(a + 1) (a - 1)}$ as a fraction with a common denominator, multiply by $\frac{a + 4}{a + 4}$ .

$\frac{1}{(a + 1) (a - 1)} \cdot \frac{a + 4}{a + 4} + \frac{1}{a + 4}$

Step 3

To write $\frac{1}{a + 4}$ as a fraction with a common denominator, multiply by $\frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$ .

$\frac{1}{(a + 1) (a - 1)} \cdot \frac{a + 4}{a + 4} + \frac{1}{a + 4} \cdot \frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$

Step 4

Write each expression with a common denominator of $(a + 1) (a - 1) (a + 4)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{1}{(a + 1) (a - 1)}$ by $\frac{a + 4}{a + 4}$ .

$\frac{a + 4}{(a + 1) (a - 1) (a + 4)} + \frac{1}{a + 4} \cdot \frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$

Step 4.2

Multiply $\frac{1}{a + 4}$ by $\frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$ .

$\frac{a + 4}{(a + 1) (a - 1) (a + 4)} + \frac{(a + 1) (a - 1)}{(a + 4) ((a + 1) (a - 1))}$

Step 4.3

Reorder the factors of $(a + 1) (a - 1) (a + 4)$ .

$\frac{a + 4}{(a + 1) (a + 4) (a - 1)} + \frac{(a + 1) (a - 1)}{(a + 4) ((a + 1) (a - 1))}$

Step 4.4

Reorder the factors of $(a + 4) ((a + 1) (a - 1))$ .

$\frac{a + 4}{(a + 1) (a + 4) (a - 1)} + \frac{(a + 1) (a - 1)}{(a + 1) (a + 4) (a - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a + 4 + (a + 1) (a - 1)}{(a + 1) (a + 4) (a - 1)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Expand $(a + 1) (a - 1)$ using the FOIL Method.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$\frac{a + 4 + a (a - 1) + 1 (a - 1)}{(a + 1) (a + 4) (a - 1)}$

Step 6.1.2

Apply the distributive property.

$\frac{a + 4 + a \cdot a + a \cdot - 1 + 1 (a - 1)}{(a + 1) (a + 4) (a - 1)}$

Step 6.1.3

Apply the distributive property.

$\frac{a + 4 + a \cdot a + a \cdot - 1 + 1 a + 1 \cdot - 1}{(a + 1) (a + 4) (a - 1)}$

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 $a$ .

$\frac{a + 4 + a^{2} + a \cdot - 1 + 1 a + 1 \cdot - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.1.2

Move $- 1$ to the left of $a$ .

$\frac{a + 4 + a^{2} - 1 \cdot a + 1 a + 1 \cdot - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.1.3

Rewrite $- 1 a$ as $- a$ .

$\frac{a + 4 + a^{2} - a + 1 a + 1 \cdot - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.1.4

Multiply $a$ by $1$ .

$\frac{a + 4 + a^{2} - a + a + 1 \cdot - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.1.5

Multiply $- 1$ by $1$ .

$\frac{a + 4 + a^{2} - a + a - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.2

Add $- a$ and $a$ .

$\frac{a + 4 + a^{2} + 0 - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.2.3

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

$\frac{a + 4 + a^{2} - 1}{(a + 1) (a + 4) (a - 1)}$

Step 6.3

Subtract $1$ from $4$ .

$\frac{a + a^{2} + 3}{(a + 1) (a + 4) (a - 1)}$

Step 6.4

Reorder terms.

$\frac{a^{2} + a + 3}{(a + 1) (a + 4) (a - 1)}$