Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Rewrite $\frac{x^{2} - {(x + h)}^{2}}{x^{2} {(x + h)}^{2}}$ in a factored form.

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Step 1.5.1

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 $b = x + h$ .

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

Step 1.5.2

Simplify.

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Step 1.5.2.1

Add $x$ and $x$ .

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

Step 1.5.2.2

Apply the distributive property.

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

Step 1.5.2.3

Subtract $x$ from $x$ .

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

Step 1.5.2.4

Subtract $h$ from $0$ .

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

Step 1.5.2.5

Factor out negative.

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

Step 1.6

Move the negative in front of the fraction.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Cancel the common factor of $h$ .

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Step 3.1

Move the leading negative in $- \frac{(2 x + h) h}{x^{2} {(x + h)}^{2}}$ into the numerator.

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

Step 3.2

Factor $h$ out of $- (2 x + h) h$ .

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

Step 3.3

Cancel the common factor.

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

Step 3.4

Rewrite the expression.

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

Step 4

Move the negative in front of the fraction.

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