Simplify x/(x^2-25)-1/(x^2-10x+25)

$\frac{x}{x^{2} - 25} - \frac{1}{x^{2} - 10 x + 25}$

Step 1

Simplify each term.

Simplify the denominator.

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

$\frac{x}{x^{2} - 5^{2}} - \frac{1}{x^{2} - 10 x + 25}$

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 $b = 5$ .

$\frac{x}{(x + 5) (x - 5)} - \frac{1}{x^{2} - 10 x + 25}$

Factor using the perfect square rule.

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

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

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.

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

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

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

Step 2

To write $\frac{x}{(x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 3

To write $- \frac{1}{{(x - 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Step 4

Write each expression with a common denominator of $(x + 5) {(x - 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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

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

Raise $x - 5$ to the power of $1$ .

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

Raise $x - 5$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

Reorder the factors of $(x + 5) {(x - 5)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $5$ .

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

Subtract $x$ from $- 5 x$ .

$\frac{x^{2} - 6 x - 5}{{(x - 5)}^{2} (x + 5)}$

Simplify x/(x^2-25)-1/(x^2-10x+25)

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