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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{25}{x}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $15$ by $- 25$ .

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

Cancel the common factor of $15$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Factor $15$ out of $x \cdot 15 x - 375$ .

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

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

Factor $15$ out of $- 375$ .

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

Factor $15$ out of $15 (x \cdot x) + 15 \cdot - 25$ .

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

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

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

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

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

Step 5

Simplify the denominator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

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.

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

Step 6

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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