Simplify (x^2-10x+25)/(10x-100)*(x-10)/(45-9x)

$\frac{x^{2} - 10 x + 25}{10 x - 100} \cdot \frac{x - 10}{45 - 9 x}$

Factor using the perfect square rule.

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

$\frac{x^{2} - 10 x + 5^{2}}{10 x - 100} \cdot \frac{x - 10}{45 - 9 x}$

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.

$\frac{x^{2} - 2 \cdot x \cdot 5 + 5^{2}}{10 x - 100} \cdot \frac{x - 10}{45 - 9 x}$

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

$\frac{{(x - 5)}^{2}}{10 x - 100} \cdot \frac{x - 10}{45 - 9 x}$

Simplify terms.

Factor $10$ out of $10 x - 100$ .

Factor $10$ out of $10 x$ .

$\frac{{(x - 5)}^{2}}{10 (x) - 100} \cdot \frac{x - 10}{45 - 9 x}$

Factor $10$ out of $- 100$ .

$\frac{{(x - 5)}^{2}}{10 x + 10 \cdot - 10} \cdot \frac{x - 10}{45 - 9 x}$

Factor $10$ out of $10 x + 10 \cdot - 10$ .

$\frac{{(x - 5)}^{2}}{10 (x - 10)} \cdot \frac{x - 10}{45 - 9 x}$

Factor $9$ out of $45 - 9 x$ .

Factor $9$ out of $45$ .

$\frac{{(x - 5)}^{2}}{10 (x - 10)} \cdot \frac{x - 10}{9 (5) - 9 x}$

Factor $9$ out of $- 9 x$ .

$\frac{{(x - 5)}^{2}}{10 (x - 10)} \cdot \frac{x - 10}{9 (5) + 9 (- x)}$

Factor $9$ out of $9 (5) + 9 (- x)$ .

$\frac{{(x - 5)}^{2}}{10 (x - 10)} \cdot \frac{x - 10}{9 (5 - x)}$

Cancel the common factor of $x - 10$ .

Factor $x - 10$ out of $10 (x - 10)$ .

$\frac{{(x - 5)}^{2}}{(x - 10) \cdot 10} \cdot \frac{x - 10}{9 (5 - x)}$

Cancel the common factor.

$\frac{{(x - 5)}^{2}}{(x - 10) \cdot 10} \cdot \frac{x - 10}{9 (5 - x)}$

Rewrite the expression.

$\frac{{(x - 5)}^{2}}{10} \cdot \frac{1}{9 (5 - x)}$

Multiply $\frac{{(x - 5)}^{2}}{10}$ by $\frac{1}{9 (5 - x)}$ .

$\frac{{(x - 5)}^{2}}{10 (9 (5 - x))}$

Multiply $9$ by $10$ .

$\frac{{(x - 5)}^{2}}{90 (5 - x)}$

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

Factor $- 1$ out of $x$ .

$\frac{{(- 1 (- x) - 5)}^{2}}{90 (5 - x)}$

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{{(- 1 (- x) - 1 (5))}^{2}}{90 (5 - x)}$

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

$\frac{{(- 1 (- x + 5))}^{2}}{90 (5 - x)}$

Apply the product rule to $- 1 (- x + 5)$ .

$\frac{{(- 1)}^{2} {(- x + 5)}^{2}}{90 (5 - x)}$

Raise $- 1$ to the power of $2$ .

$\frac{1 {(- x + 5)}^{2}}{90 (5 - x)}$

Multiply ${(- x + 5)}^{2}$ by $1$ .

$\frac{{(- x + 5)}^{2}}{90 (5 - x)}$

Reorder terms.

$\frac{{(- x + 5)}^{2}}{90 (- x + 5)}$

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

$\frac{(- x + 5) (- x + 5)}{90 (- x + 5)}$

Cancel the common factors.

Factor $- x + 5$ out of $90 (- x + 5)$ .

$\frac{(- x + 5) (- x + 5)}{(- x + 5) \cdot 90}$

Cancel the common factor.

$\frac{(- x + 5) (- x + 5)}{(- x + 5) \cdot 90}$

Rewrite the expression.

$\frac{- x + 5}{90}$

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 5}{90}$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (x) - 1 (- 5)}{90}$

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

$\frac{- (x - 5)}{90}$

Simplify the expression.

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{- 1 (x - 5)}{90}$

Move the negative in front of the fraction.

$- \frac{x - 5}{90}$

Simplify (x^2-10x+25)/(10x-100)*(x-10)/(45-9x)

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