Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Add $x$ and $x$ .

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

Step 1.2.2

Subtract $h$ from $h$ .

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

Step 1.2.3

Add $2 x$ and $0$ .

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

Step 1.2.4

Apply the distributive property.

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

Step 1.2.5

Multiply $- - h$ .

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Step 1.2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.2

Multiply $h$ by $1$ .

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

Step 1.2.6

Subtract $x$ from $x$ .

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

Step 1.2.7

Add $h$ and $0$ .

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

Step 1.2.8

Add $h$ and $h$ .

$\frac{2 x \cdot 2 h}{2 h}$

Step 1.2.9

Multiply $2$ by $2$ .

$\frac{4 x h}{2 h}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $4$ and $2$ .

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Step 2.1.1

Factor $2$ out of $4 x h$ .

$\frac{2 (2 x h)}{2 h}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $2 h$ .

$\frac{2 (2 x h)}{2 (h)}$

Step 2.1.2.2

Cancel the common factor.

$\frac{2 (2 x h)}{2 h}$

Step 2.1.2.3

Rewrite the expression.

$\frac{2 x h}{h}$

Step 2.2

Cancel the common factor of $h$ .

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Step 2.2.1

Cancel the common factor.

$\frac{2 x h}{h}$

Step 2.2.2

Divide $2 x$ by $1$ .

$2 x$