Simplify (25/x-x)/((x-8)/15+1/x)

$\frac{\frac{25}{x} - x}{\frac{x - 8}{15} + \frac{1}{x}}$

Multiply the numerator and denominator of the complex fraction by $x \cdot 15$ .

Multiply $\frac{\frac{25}{x} - x}{\frac{x - 8}{15} + \frac{1}{x}}$ by $\frac{x \cdot 15}{x \cdot 15}$ .

$\frac{x \cdot 15}{x \cdot 15} \cdot \frac{\frac{25}{x} - x}{\frac{x - 8}{15} + \frac{1}{x}}$

Combine.

$\frac{x \cdot 15 (\frac{25}{x} - x)}{x \cdot 15 (\frac{x - 8}{15} + \frac{1}{x})}$

Apply the distributive property.

$\frac{x \cdot 15 \frac{25}{x} + x \cdot 15 (- x)}{x \cdot 15 \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 15$ .

$\frac{x (15) \frac{25}{x} + x \cdot 15 (- x)}{x \cdot 15 \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Cancel the common factor.

$\frac{x \cdot 15 \frac{25}{x} + x \cdot 15 (- x)}{x \cdot 15 \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Rewrite the expression.

$\frac{15 \cdot 25 + x \cdot 15 (- x)}{x \cdot 15 \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Multiply $15$ by $25$ .

$\frac{375 + x \cdot 15 (- x)}{x \cdot 15 \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Cancel the common factor of $15$ .

Factor $15$ out of $x \cdot 15$ .

$\frac{375 + x \cdot 15 (- x)}{15 \cdot x \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Cancel the common factor.

$\frac{375 + x \cdot 15 (- x)}{15 \cdot x \frac{x - 8}{15} + x \cdot 15 \frac{1}{x}}$

Rewrite the expression.

$\frac{375 + x \cdot 15 (- x)}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 15$ .

$\frac{375 + x \cdot 15 (- x)}{x (x - 8) + x (15) \frac{1}{x}}$

Cancel the common factor.

$\frac{375 + x \cdot 15 (- x)}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Rewrite the expression.

$\frac{375 + x \cdot 15 (- x)}{x (x - 8) + 15}$

Simplify the numerator.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{375 + x \cdot x \cdot 15 \cdot - 1}{x (x - 8) + 15}$

Multiply $x$ by $x$ .

$\frac{375 + x^{2} \cdot 15 \cdot - 1}{x (x - 8) + 15}$

Move $15$ to the left of $x^{2}$ .

$\frac{375 + 15 \cdot x^{2} \cdot - 1}{x (x - 8) + 15}$

Multiply $- 1$ by $15$ .

$\frac{375 - 15 x^{2}}{x (x - 8) + 15}$

Rewrite $375 - 15 x^{2}$ in a factored form.

Factor $15$ out of $375 - 15 x^{2}$ .

Factor $15$ out of $375$ .

$\frac{15 (25) - 15 x^{2}}{x (x - 8) + 15}$

Factor $15$ out of $- 15 x^{2}$ .

$\frac{15 (25) + 15 (- x^{2})}{x (x - 8) + 15}$

Factor $15$ out of $15 (25) + 15 (- x^{2})$ .

$\frac{15 (25 - x^{2})}{x (x - 8) + 15}$

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

$\frac{15 (5^{2} - x^{2})}{x (x - 8) + 15}$

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

$\frac{15 (5 + x) (5 - x)}{x (x - 8) + 15}$

Simplify the denominator.

Apply the distributive property.

$\frac{15 (5 + x) (5 - x)}{x \cdot x + x \cdot - 8 + 15}$

Multiply $x$ by $x$ .

$\frac{15 (5 + x) (5 - x)}{x^{2} + x \cdot - 8 + 15}$

Move $- 8$ to the left of $x$ .

$\frac{15 (5 + x) (5 - x)}{x^{2} - 8 \cdot x + 15}$

Factor $x^{2} - 8 x + 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$\frac{15 (5 + x) (5 - x)}{(x - 5) (x - 3)}$

Reduce the expression by cancelling the common factors.

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

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

$\frac{15 (5 + x) (- 1 (- 5) - x)}{(x - 5) (x - 3)}$

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

$\frac{15 (5 + x) (- 1 (- 5) - (x))}{(x - 5) (x - 3)}$

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

$\frac{15 (5 + x) (- 1 (- 5 + x))}{(x - 5) (x - 3)}$

Reorder terms.

$\frac{15 (5 + x) (- 1 (x - 5))}{(x - 5) (x - 3)}$

Cancel the common factor.

$\frac{15 (5 + x) (- 1 (x - 5))}{(x - 5) (x - 3)}$

Rewrite the expression.

$\frac{15 (5 + x) \cdot (- 1)}{x - 3}$

Simplify the expression.

Multiply $- 1$ by $15$ .

$\frac{- 15 (5 + x)}{x - 3}$

Move the negative in front of the fraction.

$- \frac{15 (5 + x)}{x - 3}$

Simplify (25/x-x)/((x-8)/15+1/x)

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