Simplify ((x^2-2x)/(x^2-10x+25))÷((6x^2-12x)/(x^2-25))

$\frac{x^{2} - 2 x}{x^{2} - 10 x + 25} \div \frac{6 x^{2} - 12 x}{x^{2} - 25}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 2 x}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

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

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

$\frac{x \cdot x - 2 x}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

Factor $x$ out of $- 2 x$ .

$\frac{x \cdot x + x \cdot - 2}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$\frac{x (x - 2)}{x^{2} - 10 x + 25} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$\frac{x (x - 2)}{x^{2} - 10 x + 5^{2}} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

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

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

$\frac{x (x - 2)}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{x^{2} - 25}{6 x^{2} - 12 x}$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{x^{2} - 5^{2}}{6 x^{2} - 12 x}$

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 $b = 5$ .

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} - 12 x}$

Simplify terms.

Factor $6 x$ out of $6 x^{2} - 12 x$ .

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

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{(x + 5) (x - 5)}{6 x (x) - 12 x}$

Factor $6 x$ out of $- 12 x$ .

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{(x + 5) (x - 5)}{6 x (x) + 6 x (- 2)}$

Factor $6 x$ out of $6 x (x) + 6 x (- 2)$ .

$\frac{x (x - 2)}{{(x - 5)}^{2}} \cdot \frac{(x + 5) (x - 5)}{6 x (x - 2)}$

Combine.

$\frac{x (x - 2) ((x + 5) (x - 5))}{{(x - 5)}^{2} (6 x (x - 2))}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x - 2) ((x + 5) (x - 5))}{{(x - 5)}^{2} (6 x (x - 2))}$

Rewrite the expression.

$\frac{(x - 2) ((x + 5) (x - 5))}{{(x - 5)}^{2} (6 (x - 2))}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{(x - 2) ((x + 5) (x - 5))}{{(x - 5)}^{2} (6 (x - 2))}$

Rewrite the expression.

$\frac{(x + 5) (x - 5)}{{(x - 5)}^{2} \cdot (6)}$

Cancel the common factor of $x - 5$ and ${(x - 5)}^{2}$ .

Factor $x - 5$ out of $(x + 5) (x - 5)$ .

$\frac{(x - 5) (x + 5)}{{(x - 5)}^{2} \cdot (6)}$

Cancel the common factors.

Factor $x - 5$ out of ${(x - 5)}^{2} \cdot (6)$ .

$\frac{(x - 5) (x + 5)}{(x - 5) ((x - 5) \cdot 6)}$

Cancel the common factor.

$\frac{(x - 5) (x + 5)}{(x - 5) ((x - 5) \cdot 6)}$

Rewrite the expression.

$\frac{x + 5}{(x - 5) \cdot 6}$

Move $6$ to the left of $x - 5$ .

$\frac{x + 5}{6 (x - 5)}$

Simplify ((x^2-2x)/(x^2-10x+25))÷((6x^2-12x)/(x^2-25))

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