Algebra Examples

$\frac{2}{{(x + h)}^{2} - 3} - \frac{2}{x^{2} - 3}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$\frac{2}{(x + h) (x + h) - 3} - \frac{2}{x^{2} - 3}$

Step 1.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{2}{x (x + h) + h (x + h) - 3} - \frac{2}{x^{2} - 3}$

Step 1.2.2

Apply the distributive property.

$\frac{2}{x \cdot x + x h + h (x + h) - 3} - \frac{2}{x^{2} - 3}$

Step 1.2.3

Apply the distributive property.

$\frac{2}{x \cdot x + x h + h x + h \cdot h - 3} - \frac{2}{x^{2} - 3}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\frac{2}{x^{2} + x h + h x + h \cdot h - 3} - \frac{2}{x^{2} - 3}$

Step 1.3.1.2

Multiply $h$ by $h$ .

$\frac{2}{x^{2} + x h + h x + h^{2} - 3} - \frac{2}{x^{2} - 3}$

Step 1.3.2

Add $x h$ and $h x$ .

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Step 1.3.2.1

Reorder $x$ and $h$ .

$\frac{2}{x^{2} + h x + h x + h^{2} - 3} - \frac{2}{x^{2} - 3}$

Step 1.3.2.2

Add $h x$ and $h x$ .

$\frac{2}{x^{2} + 2 h x + h^{2} - 3} - \frac{2}{x^{2} - 3}$

Step 2

To write $\frac{2}{x^{2} + 2 h x + h^{2} - 3}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 3}{x^{2} - 3}$ .

$\frac{2}{x^{2} + 2 h x + h^{2} - 3} \cdot \frac{x^{2} - 3}{x^{2} - 3} - \frac{2}{x^{2} - 3}$

Step 3

To write $- \frac{2}{x^{2} - 3}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 2 h x + h^{2} - 3}{x^{2} + 2 h x + h^{2} - 3}$ .

$\frac{2}{x^{2} + 2 h x + h^{2} - 3} \cdot \frac{x^{2} - 3}{x^{2} - 3} - \frac{2}{x^{2} - 3} \cdot \frac{x^{2} + 2 h x + h^{2} - 3}{x^{2} + 2 h x + h^{2} - 3}$

Step 4

Write each expression with a common denominator of $(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{x^{2} + 2 h x + h^{2} - 3}$ by $\frac{x^{2} - 3}{x^{2} - 3}$ .

$\frac{2 (x^{2} - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)} - \frac{2}{x^{2} - 3} \cdot \frac{x^{2} + 2 h x + h^{2} - 3}{x^{2} + 2 h x + h^{2} - 3}$

Step 4.2

Multiply $\frac{2}{x^{2} - 3}$ by $\frac{x^{2} + 2 h x + h^{2} - 3}{x^{2} + 2 h x + h^{2} - 3}$ .

$\frac{2 (x^{2} - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)} - \frac{2 (x^{2} + 2 h x + h^{2} - 3)}{(x^{2} - 3) (x^{2} + 2 h x + h^{2} - 3)}$

Step 4.3

Reorder the factors of $(x^{2} - 3) (x^{2} + 2 h x + h^{2} - 3)$ .

$\frac{2 (x^{2} - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)} - \frac{2 (x^{2} + 2 h x + h^{2} - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (x^{2} - 3) - 2 (x^{2} + 2 h x + h^{2} - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 (x^{2} - 3) - 2 (x^{2} + 2 h x + h^{2} - 3)$ .

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Step 6.1.1

Factor $2$ out of $- 2 (x^{2} + 2 h x + h^{2} - 3)$ .

$\frac{2 (x^{2} - 3) + 2 (- (x^{2} + 2 h x + h^{2} - 3))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.1.2

Factor $2$ out of $2 (x^{2} - 3) + 2 (- (x^{2} + 2 h x + h^{2} - 3))$ .

$\frac{2 (x^{2} - 3 - (x^{2} + 2 h x + h^{2} - 3))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.2

Apply the distributive property.

$\frac{2 (x^{2} - 3 - x^{2} - (2 h x) - h^{2} - - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} - 3 - x^{2} - 2 (h x) - h^{2} - - 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.3.2

Multiply $- 1$ by $- 3$ .

$\frac{2 (x^{2} - 3 - x^{2} - 2 (h x) - h^{2} + 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.4

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

$\frac{2 (0 - 3 - 2 h x - h^{2} + 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.5

Subtract $3$ from $0$ .

$\frac{2 (- 3 - 2 h x - h^{2} + 3)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.6

Add $- 3$ and $3$ .

$\frac{2 (- 2 h x - h^{2} + 0)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.7

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

$\frac{2 (- 2 h x - h^{2})}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.8

Factor $h$ out of $- 2 h x - h^{2}$ .

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Step 6.8.1

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

$\frac{2 (h (- 2 x) - h^{2})}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.8.2

Factor $h$ out of $- h^{2}$ .

$\frac{2 (h (- 2 x) + h (- h))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 6.8.3

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

$\frac{2 (h (- 2 x - h))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

$\frac{2 h (- 2 x - h)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 2 x$ .

$\frac{2 h (- (2 x) - h)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7.2

Factor $- 1$ out of $- h$ .

$\frac{2 h (- (2 x) - (h))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7.3

Factor $- 1$ out of $- (2 x) - (h)$ .

$\frac{2 h (- (2 x + h))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (2 x + h)$ as $- 1 (2 x + h)$ .

$\frac{2 h (- 1 (2 x + h))}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{(2 h) (2 x + h)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$

Step 7.4.3

Reorder factors in $- \frac{(2 h) (2 x + h)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$ .

$- \frac{2 h (2 x + h)}{(x^{2} + 2 h x + h^{2} - 3) (x^{2} - 3)}$