Simplify (x-5)/(10x-2)*(25x^2-1)/(x^2-10x+25)

$\frac{x - 5}{10 x - 2} \cdot \frac{25 x^{2} - 1}{x^{2} - 10 x + 25}$

Step 1

Factor $2$ out of $10 x - 2$ .

Factor $2$ out of $10 x$ .

$\frac{x - 5}{2 (5 x) - 2} \cdot \frac{25 x^{2} - 1}{x^{2} - 10 x + 25}$

Factor $2$ out of $- 2$ .

$\frac{x - 5}{2 (5 x) + 2 (- 1)} \cdot \frac{25 x^{2} - 1}{x^{2} - 10 x + 25}$

Factor $2$ out of $2 (5 x) + 2 (- 1)$ .

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{25 x^{2} - 1}{x^{2} - 10 x + 25}$

Step 2

Simplify the numerator.

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{{(5 x)}^{2} - 1}{x^{2} - 10 x + 25}$

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

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{{(5 x)}^{2} - 1^{2}}{x^{2} - 10 x + 25}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 1$ .

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{x^{2} - 10 x + 25}$

Step 3

Factor using the perfect square rule.

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

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{x^{2} - 10 x + 5^{2}}$

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.

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}}$

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

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{{(x - 5)}^{2}}$

Step 4

Simplify terms.

Cancel the common factor of $x - 5$ .

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

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{(x - 5) (x - 5)}$

Cancel the common factor.

$\frac{x - 5}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{(x - 5) (x - 5)}$

Rewrite the expression.

$\frac{1}{2 (5 x - 1)} \cdot \frac{(5 x + 1) (5 x - 1)}{x - 5}$

Cancel the common factor of $5 x - 1$ .

Factor $5 x - 1$ out of $2 (5 x - 1)$ .

$\frac{1}{(5 x - 1) \cdot 2} \cdot \frac{(5 x + 1) (5 x - 1)}{x - 5}$

Factor $5 x - 1$ out of $(5 x + 1) (5 x - 1)$ .

$\frac{1}{(5 x - 1) \cdot 2} \cdot \frac{(5 x - 1) (5 x + 1)}{x - 5}$

Cancel the common factor.

$\frac{1}{(5 x - 1) \cdot 2} \cdot \frac{(5 x - 1) (5 x + 1)}{x - 5}$

Rewrite the expression.

$\frac{1}{2} \cdot \frac{5 x + 1}{x - 5}$

Multiply $\frac{1}{2}$ by $\frac{5 x + 1}{x - 5}$ .

$\frac{5 x + 1}{2 (x - 5)}$

Simplify (x-5)/(10x-2)*(25x^2-1)/(x^2-10x+25)

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