Calculus Examples

$\frac{6 - 3 (2 + h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $6 - 3 (2 + h)$ .

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Step 1.1.1

Factor $3$ out of $6$ .

$\frac{3 (2) - 3 (2 + h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.1.2

Factor $3$ out of $- 3 (2 + h)$ .

$\frac{3 (2) + 3 (- (2 + h))}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.1.3

Factor $3$ out of $3 (2) + 3 (- (2 + h))$ .

$\frac{3 (2 - (2 + h))}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.2

Apply the distributive property.

$\frac{3 (2 - 1 \cdot 2 - h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.3

Multiply $- 1$ by $2$ .

$\frac{3 (2 - 2 - h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.4

Subtract $2$ from $2$ .

$\frac{3 (0 - h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.5

Subtract $h$ from $0$ .

$\frac{3 \cdot - 1 h}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.6

Combine exponents.

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Step 1.6.1

Factor out negative.

$\frac{- (3 h)}{\sqrt{8 + 4 h + 1} - 3}$

Step 1.6.2

Multiply $3$ by $- 1$ .

$\frac{- 3 h}{\sqrt{8 + 4 h + 1} - 3}$

Step 2

Simplify the expression.

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Step 2.1

Add $8$ and $1$ .

$\frac{- 3 h}{\sqrt{4 h + 9} - 3}$

Step 2.2

Move the negative in front of the fraction.

$- \frac{3 h}{\sqrt{4 h + 9} - 3}$

Step 3

Multiply $\frac{3 h}{\sqrt{4 h + 9} - 3}$ by $\frac{\sqrt{4 h + 9} + 3}{\sqrt{4 h + 9} + 3}$ .

$- (\frac{3 h}{\sqrt{4 h + 9} - 3} \cdot \frac{\sqrt{4 h + 9} + 3}{\sqrt{4 h + 9} + 3})$

Step 4

Multiply $\frac{3 h}{\sqrt{4 h + 9} - 3}$ by $\frac{\sqrt{4 h + 9} + 3}{\sqrt{4 h + 9} + 3}$ .

$- \frac{3 h (\sqrt{4 h + 9} + 3)}{(\sqrt{4 h + 9} - 3) (\sqrt{4 h + 9} + 3)}$

Step 5

Expand the denominator using the FOIL method.

$- \frac{3 h (\sqrt{4 h + 9} + 3)}{{\sqrt{4 h + 9}}^{2} + \sqrt{4 h + 9} \cdot 3 - 3 \sqrt{4 h + 9} - 9}$

Step 6

Simplify.

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Step 6.1

Subtract $9$ from $9$ .

$- \frac{3 h (\sqrt{4 h + 9} + 3)}{4 h + 0}$

Step 6.2

Add $4 h$ and $0$ .

$- \frac{3 h (\sqrt{4 h + 9} + 3)}{4 h}$

Step 7

Cancel the common factor of $h$ .

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Step 7.1

Cancel the common factor.

$- \frac{3 h (\sqrt{4 h + 9} + 3)}{4 h}$

Step 7.2

Rewrite the expression.

$- \frac{3 (\sqrt{4 h + 9} + 3)}{4}$