Simplify (1-(x^2)/4)/((x-3)/10-1/x)

$\frac{1 - \frac{x^{2}}{4}}{\frac{x - 3}{10} - \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $20 x$ .

Multiply $\frac{1 - \frac{x^{2}}{4}}{\frac{x - 3}{10} - \frac{1}{x}}$ by $\frac{20 x}{20 x}$ .

$\frac{20 x}{20 x} \cdot \frac{1 - \frac{x^{2}}{4}}{\frac{x - 3}{10} - \frac{1}{x}}$

Combine.

$\frac{20 x (1 - \frac{x^{2}}{4})}{20 x (\frac{x - 3}{10} - \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{20 x \cdot 1 + 20 x (- \frac{x^{2}}{4})}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$\frac{20 x \cdot 1 + 20 x \frac{- x^{2}}{4}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Factor $4$ out of $20 x$ .

$\frac{20 x \cdot 1 + 4 (5 x) \frac{- x^{2}}{4}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Cancel the common factor.

$\frac{20 x \cdot 1 + 4 (5 x) \frac{- x^{2}}{4}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Rewrite the expression.

$\frac{20 x \cdot 1 + 5 x (- x^{2})}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Multiply $- 1$ by $5$ .

$\frac{20 x \cdot 1 - 5 x \cdot x^{2}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{20 x \cdot 1 - 5 (x^{2} x)}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Multiply $x^{2}$ by $x$ .

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

$\frac{20 x \cdot 1 - 5 (x^{2} x^{1})}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

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

$\frac{20 x \cdot 1 - 5 x^{2 + 1}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Add $2$ and $1$ .

$\frac{20 x \cdot 1 - 5 x^{3}}{20 x \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Cancel the common factor of $10$ .

Factor $10$ out of $20 x$ .

$\frac{20 x \cdot 1 - 5 x^{3}}{10 (2 x) \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Cancel the common factor.

$\frac{20 x \cdot 1 - 5 x^{3}}{10 (2 x) \frac{x - 3}{10} + 20 x (- \frac{1}{x})}$

Rewrite the expression.

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) + 20 x (- \frac{1}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) + 20 x \frac{- 1}{x}}$

Factor $x$ out of $20 x$ .

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) + x \cdot 20 \frac{- 1}{x}}$

Cancel the common factor.

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) + x \cdot 20 \frac{- 1}{x}}$

Rewrite the expression.

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) + 20 \cdot - 1}$

Multiply $20$ by $- 1$ .

$\frac{20 x \cdot 1 - 5 x^{3}}{2 x (x - 3) - 20}$

Step 4

Simplify the numerator.

Factor $5 x$ out of $20 x \cdot 1 - 5 x^{3}$ .

Factor $5 x$ out of $20 x \cdot 1$ .

$\frac{5 x (4 \cdot 1) - 5 x^{3}}{2 x (x - 3) - 20}$

Factor $5 x$ out of $- 5 x^{3}$ .

$\frac{5 x (4 \cdot 1) + 5 x (- x^{2})}{2 x (x - 3) - 20}$

Factor $5 x$ out of $5 x (4 \cdot 1) + 5 x (- x^{2})$ .

$\frac{5 x (4 \cdot 1 - x^{2})}{2 x (x - 3) - 20}$

Multiply $4$ by $1$ .

$\frac{5 x (4 - x^{2})}{2 x (x - 3) - 20}$

Rewrite $4$ as $2^{2}$ .

$\frac{5 x (2^{2} - x^{2})}{2 x (x - 3) - 20}$

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

$\frac{5 x (2 + x) (2 - x)}{2 x (x - 3) - 20}$

Step 5

Simplify the denominator.

Factor $2$ out of $2 x (x - 3) - 20$ .

Factor $2$ out of $2 x (x - 3)$ .

$\frac{5 x (2 + x) (2 - x)}{2 (x (x - 3)) - 20}$

Factor $2$ out of $- 20$ .

$\frac{5 x (2 + x) (2 - x)}{2 (x (x - 3)) + 2 \cdot - 10}$

Factor $2$ out of $2 (x (x - 3)) + 2 \cdot - 10$ .

$\frac{5 x (2 + x) (2 - x)}{2 (x (x - 3) - 10)}$

Apply the distributive property.

$\frac{5 x (2 + x) (2 - x)}{2 (x \cdot x + x \cdot - 3 - 10)}$

Multiply $x$ by $x$ .

$\frac{5 x (2 + x) (2 - x)}{2 (x^{2} + x \cdot - 3 - 10)}$

Move $- 3$ to the left of $x$ .

$\frac{5 x (2 + x) (2 - x)}{2 (x^{2} - 3 \cdot x - 10)}$

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $2 + x$ and $x + 2$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1-(x^2)/4)/((x-3)/10-1/x)

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