Algebra Examples

$\frac{1}{12 a - 3} - \frac{16 a^{2}}{12 a - 3}$

Step 1

Combine the numerators over the common denominator.

$\frac{1 - 16 a^{2}}{12 a - 3}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{1^{2} - 16 a^{2}}{12 a - 3}$

Step 2.2

Rewrite $16 a^{2}$ as ${(4 a)}^{2}$ .

$\frac{1^{2} - {(4 a)}^{2}}{12 a - 3}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 4 a$ .

$\frac{(1 + 4 a) (1 - (4 a))}{12 a - 3}$

Step 2.4

Multiply $4$ by $- 1$ .

$\frac{(1 + 4 a) (1 - 4 a)}{12 a - 3}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $12 a - 3$ .

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Step 3.1.1

Factor $3$ out of $12 a$ .

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

Step 3.1.2

Factor $3$ out of $- 3$ .

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

Step 3.1.3

Factor $3$ out of $3 (4 a) + 3 (- 1)$ .

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

Step 3.2

Cancel the common factor of $1 - 4 a$ and $4 a - 1$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $- 1$ out of $- 4 a$ .

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

Step 3.2.3

Factor $- 1$ out of $- 1 (- 1) - (4 a)$ .

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

Step 3.2.4

Reorder terms.

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

Step 3.2.5

Cancel the common factor.

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

Step 3.2.6

Rewrite the expression.

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Move $- 1$ to the left of $1 + 4 a$ .

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

Step 3.3.2

Move the negative in front of the fraction.

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