Simplify (3x^3-12x)/(6x^3-24x^2+24x)

$\frac{3 x^{3} - 12 x}{6 x^{3} - 24 x^{2} + 24 x}$

Simplify the numerator.

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

Factor $3 x$ out of $3 x^{3}$ .

$\frac{3 x (x^{2}) - 12 x}{6 x^{3} - 24 x^{2} + 24 x}$

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

$\frac{3 x (x^{2}) + 3 x (- 4)}{6 x^{3} - 24 x^{2} + 24 x}$

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

$\frac{3 x (x^{2} - 4)}{6 x^{3} - 24 x^{2} + 24 x}$

Rewrite $4$ as $2^{2}$ .

$\frac{3 x (x^{2} - 2^{2})}{6 x^{3} - 24 x^{2} + 24 x}$

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

$\frac{3 x (x + 2) (x - 2)}{6 x^{3} - 24 x^{2} + 24 x}$

Simplify the denominator.

Factor $6 x$ out of $6 x^{3} - 24 x^{2} + 24 x$ .

Factor $6 x$ out of $6 x^{3}$ .

$\frac{3 x (x + 2) (x - 2)}{6 x (x^{2}) - 24 x^{2} + 24 x}$

Factor $6 x$ out of $- 24 x^{2}$ .

$\frac{3 x (x + 2) (x - 2)}{6 x (x^{2}) + 6 x (- 4 x) + 24 x}$

Factor $6 x$ out of $24 x$ .

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

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 4 x)$ .

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

Factor $6 x$ out of $6 x (x^{2} - 4 x) + 6 x (4)$ .

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

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

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

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$\frac{3 x (x + 2) (x - 2)}{6 x (x^{2} - 2 \cdot x \cdot 2 + 2^{2})}$

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

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

Reduce the expression by cancelling the common factors.

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

Factor $3$ out of $3 x (x + 2) (x - 2)$ .

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

Cancel the common factors.

Factor $3$ out of $6 x {(x - 2)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x (x + 2)) (x - 2)}{2 x {(x - 2)}^{2}}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x + 2) (x - 2)}{2 x {(x - 2)}^{2}}$

Rewrite the expression.

$\frac{(x + 2) (x - 2)}{2 {(x - 2)}^{2}}$

Cancel the common factor of $x - 2$ and ${(x - 2)}^{2}$ .

Factor $x - 2$ out of $(x + 2) (x - 2)$ .

$\frac{(x - 2) (x + 2)}{2 {(x - 2)}^{2}}$

Cancel the common factors.

Factor $x - 2$ out of $2 {(x - 2)}^{2}$ .

$\frac{(x - 2) (x + 2)}{(x - 2) (2 (x - 2))}$

Cancel the common factor.

$\frac{(x - 2) (x + 2)}{(x - 2) (2 (x - 2))}$

Rewrite the expression.

$\frac{x + 2}{2 (x - 2)}$

Simplify (3x^3-12x)/(6x^3-24x^2+24x)

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