Simplify (12a^3-3a)/(12a^3+6a^2)

$\frac{12 a^{3} - 3 a}{12 a^{3} + 6 a^{2}}$

Step 1

Simplify the numerator.

Factor $3 a$ out of $12 a^{3} - 3 a$ .

Factor $3 a$ out of $12 a^{3}$ .

$\frac{3 a (4 a^{2}) - 3 a}{12 a^{3} + 6 a^{2}}$

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

$\frac{3 a (4 a^{2}) + 3 a (- 1)}{12 a^{3} + 6 a^{2}}$

Factor $3 a$ out of $3 a (4 a^{2}) + 3 a (- 1)$ .

$\frac{3 a (4 a^{2} - 1)}{12 a^{3} + 6 a^{2}}$

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

$\frac{3 a ({(2 a)}^{2} - 1)}{12 a^{3} + 6 a^{2}}$

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

$\frac{3 a ({(2 a)}^{2} - 1^{2})}{12 a^{3} + 6 a^{2}}$

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

$\frac{3 a (2 a + 1) (2 a - 1)}{12 a^{3} + 6 a^{2}}$

Step 2

Simplify terms.

Factor $6 a^{2}$ out of $12 a^{3} + 6 a^{2}$ .

Factor $6 a^{2}$ out of $12 a^{3}$ .

$\frac{3 a (2 a + 1) (2 a - 1)}{6 a^{2} (2 a) + 6 a^{2}}$

Factor $6 a^{2}$ out of $6 a^{2}$ .

$\frac{3 a (2 a + 1) (2 a - 1)}{6 a^{2} (2 a) + 6 a^{2} (1)}$

Factor $6 a^{2}$ out of $6 a^{2} (2 a) + 6 a^{2} (1)$ .

$\frac{3 a (2 a + 1) (2 a - 1)}{6 a^{2} (2 a + 1)}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 a (2 a + 1) (2 a - 1)$ .

$\frac{3 ((a (2 a + 1)) (2 a - 1))}{6 a^{2} (2 a + 1)}$

Cancel the common factors.

Factor $3$ out of $6 a^{2} (2 a + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $a$ and $a^{2}$ .

Factor $a$ out of $(a (2 a + 1)) (2 a - 1)$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 a + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 a - 1}{2 a}$

Simplify (12a^3-3a)/(12a^3+6a^2)

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