Simplify ((x-2)/(x+5)+(x-5)/(x+2))/((x-2)/(x+5)-(x-5)/(x+2))

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

Step 1

Multiply the numerator and denominator of the complex fraction by $(x + 5) (x + 2)$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Move the leading negative in $- \frac{x - 5}{x + 2}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Expand $(x + 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Add $- 2 x$ and $2 x$ .

$\frac{x \cdot x + 0 + 2 \cdot - 2 + (x + 5) (x - 5)}{(x + 2) (x - 2) + (x + 5) (- (x - 5))}$

Add $x \cdot x$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 2$ .

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

Expand $(x + 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

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

Add $- 5 x$ and $5 x$ .

$\frac{x^{2} - 4 + x \cdot x + 0 + 5 \cdot - 5}{(x + 2) (x - 2) + (x + 5) (- (x - 5))}$

Add $x \cdot x$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $5$ by $- 5$ .

$\frac{x^{2} - 4 + x^{2} - 25}{(x + 2) (x - 2) + (x + 5) (- (x - 5))}$

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x^{2} - 4 - 25}{(x + 2) (x - 2) + (x + 5) (- (x - 5))}$

Subtract $25$ from $- 4$ .

$\frac{2 x^{2} - 29}{(x + 2) (x - 2) + (x + 5) (- (x - 5))}$

Step 5

Simplify the denominator.

Expand $(x + 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x^{2} - 29}{x (x - 2) + 2 (x - 2) + (x + 5) (- (x - 5))}$

Apply the distributive property.

$\frac{2 x^{2} - 29}{x \cdot x + x \cdot - 2 + 2 (x - 2) + (x + 5) (- (x - 5))}$

Apply the distributive property.

$\frac{2 x^{2} - 29}{x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2 + (x + 5) (- (x - 5))}$

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$\frac{2 x^{2} - 29}{x \cdot x - 2 x + 2 x + 2 \cdot - 2 + (x + 5) (- (x - 5))}$

Add $- 2 x$ and $2 x$ .

$\frac{2 x^{2} - 29}{x \cdot x + 0 + 2 \cdot - 2 + (x + 5) (- (x - 5))}$

Add $x \cdot x$ and $0$ .

$\frac{2 x^{2} - 29}{x \cdot x + 2 \cdot - 2 + (x + 5) (- (x - 5))}$

Simplify each term.

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 29}{x^{2} + 2 \cdot - 2 + (x + 5) (- (x - 5))}$

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 + (x + 5) (- (x - 5))}$

Apply the distributive property.

$\frac{2 x^{2} - 29}{x^{2} - 4 + (x + 5) (- x - - 5)}$

Multiply $- 1$ by $- 5$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 + (x + 5) (- x + 5)}$

Expand $(x + 5) (- x + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x^{2} - 29}{x^{2} - 4 + x (- x + 5) + 5 (- x + 5)}$

Apply the distributive property.

$\frac{2 x^{2} - 29}{x^{2} - 4 + x (- x) + x \cdot 5 + 5 (- x + 5)}$

Apply the distributive property.

$\frac{2 x^{2} - 29}{x^{2} - 4 + x (- x) + x \cdot 5 + 5 (- x) + 5 \cdot 5}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - 29}{x^{2} - 4 - x \cdot x + x \cdot 5 + 5 (- x) + 5 \cdot 5}$

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

Move $x$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - (x \cdot x) + x \cdot 5 + 5 (- x) + 5 \cdot 5}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + x \cdot 5 + 5 (- x) + 5 \cdot 5}$

Move $5$ to the left of $x$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + 5 \cdot x + 5 (- x) + 5 \cdot 5}$

Multiply $- 1$ by $5$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + 5 x - 5 x + 5 \cdot 5}$

Multiply $5$ by $5$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + 5 x - 5 x + 25}$

Subtract $5 x$ from $5 x$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + 0 + 25}$

Add $- x^{2}$ and $0$ .

$\frac{2 x^{2} - 29}{x^{2} - 4 - x^{2} + 25}$

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 x^{2} - 29}{0 - 4 + 25}$

Subtract $4$ from $0$ .

$\frac{2 x^{2} - 29}{- 4 + 25}$

Add $- 4$ and $25$ .

$\frac{2 x^{2} - 29}{21}$

Simplify ((x-2)/(x+5)+(x-5)/(x+2))/((x-2)/(x+5)-(x-5)/(x+2))

Related Questions

Support LCMGCF.com by disabling your adblocker.