Simplify 1/((x+h)^2)-1/(x^2)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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 $b = x + h$ .

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

Simplify.

Add $x$ and $x$ .

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

Apply the distributive property.

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

Subtract $x$ from $x$ .

$\frac{(2 x + h) (0 - h)}{x^{2} {(x + h)}^{2}}$

Subtract $h$ from $0$ .

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

Factor out negative.

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

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

Step 6

Simplify the expression.

Move the negative in front of the fraction.

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

Reorder factors in $- \frac{(2 x + h) h}{x^{2} {(x + h)}^{2}}$ .

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

Simplify 1/((x+h)^2)-1/(x^2)

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