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

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

Step 1

Cancel the common factor of $3 x^{2} - 12 x + 12$ and $6 x^{2} - 24$ .

Factor $3$ out of $3 x^{2}$ .

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

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

$\frac{3 (x^{2}) + 3 (- 4 x) + 12}{6 x^{2} - 24}$

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

$\frac{3 (x^{2} - 4 x) + 12}{6 x^{2} - 24}$

Factor $3$ out of $12$ .

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

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

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

Cancel the common factors.

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

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

Factor $3$ out of $- 24$ .

$\frac{3 (x^{2} - 4 x + 4)}{3 (2 x^{2}) + 3 (- 8)}$

Factor $3$ out of $3 (2 x^{2}) + 3 (- 8)$ .

$\frac{3 (x^{2} - 4 x + 4)}{3 (2 x^{2} - 8)}$

Cancel the common factor.

$\frac{3 (x^{2} - 4 x + 4)}{3 (2 x^{2} - 8)}$

Rewrite the expression.

$\frac{x^{2} - 4 x + 4}{2 x^{2} - 8}$

Step 2

Factor using the perfect square rule.

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

$\frac{x^{2} - 4 x + 2^{2}}{2 x^{2} - 8}$

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.

$\frac{x^{2} - 2 \cdot x \cdot 2 + 2^{2}}{2 x^{2} - 8}$

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

$\frac{{(x - 2)}^{2}}{2 x^{2} - 8}$

Step 3

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 8$ .

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

$\frac{{(x - 2)}^{2}}{2 (x^{2}) - 8}$

Factor $2$ out of $- 8$ .

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

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

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

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

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

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 $b = 2$ .

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

Step 4

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

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

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

Cancel the common factors.

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

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

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

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