Precalculus Examples

$f (x) = - \frac{1}{8} {(x + 1)}^{2} + 8$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - \frac{1}{8} \cdot {((x + h) + 1)}^{2} + 8$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite ${(x + h + 1)}^{2}$ as $(x + h + 1) (x + h + 1)$ .

$f (x + h) = - \frac{1}{8} \cdot ((x + h + 1) (x + h + 1)) + 8$

Step 2.1.2.1.2

Expand $(x + h + 1) (x + h + 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x + h) = - \frac{1}{8} \cdot (x \cdot x + x h + x \cdot 1 + h x + h \cdot h + h \cdot 1 + 1 x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3

Simplify each term.

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Step 2.1.2.1.3.1

Multiply $x$ by $x$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x \cdot 1 + h x + h \cdot h + h \cdot 1 + 1 x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.2

Multiply $x$ by $1$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h \cdot h + h \cdot 1 + 1 x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.3

Multiply $h$ by $h$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h^{2} + h \cdot 1 + 1 x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.4

Multiply $h$ by $1$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h^{2} + h + 1 x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.5

Multiply $x$ by $1$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h^{2} + h + x + 1 h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.6

Multiply $h$ by $1$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h^{2} + h + x + h + 1 \cdot 1) + 8$

Step 2.1.2.1.3.7

Multiply $1$ by $1$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + x h + x + h x + h^{2} + h + x + h + 1) + 8$

Step 2.1.2.1.4

Add $x h$ and $h x$ .

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Step 2.1.2.1.4.1

Reorder $x$ and $h$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + h x + h x + x + h^{2} + h + x + h + 1) + 8$

Step 2.1.2.1.4.2

Add $h x$ and $h x$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + 2 h x + x + h^{2} + h + x + h + 1) + 8$

Step 2.1.2.1.5

Add $x$ and $x$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + 2 h x + h^{2} + h + 2 x + h + 1) + 8$

Step 2.1.2.1.6

Add $h$ and $h$ .

$f (x + h) = - \frac{1}{8} \cdot (x^{2} + 2 h x + h^{2} + 2 h + 2 x + 1) + 8$

Step 2.1.2.1.7

Apply the distributive property.

$f (x + h) = - \frac{1}{8} x^{2} - \frac{1}{8} \cdot (2 h x) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8

Simplify.

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Step 2.1.2.1.8.1

Combine $x^{2}$ and $\frac{1}{8}$ .

$f (x + h) = - \frac{x^{2}}{8} - \frac{1}{8} \cdot (2 h x) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.2

Cancel the common factor of $2$ .

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Step 2.1.2.1.8.2.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$f (x + h) = - \frac{x^{2}}{8} + \frac{- 1}{8} \cdot (2 h x) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.2.2

Factor $2$ out of $8$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{- 1}{2 (4)} \cdot (2 h x) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.2.3

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

$f (x + h) = - \frac{x^{2}}{8} + \frac{- 1}{2 (4)} \cdot (2 (h x)) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.2.4

Cancel the common factor.

$f (x + h) = - \frac{x^{2}}{8} + \frac{- 1}{2 \cdot 4} \cdot (2 (h x)) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.2.5

Rewrite the expression.

$f (x + h) = - \frac{x^{2}}{8} + \frac{- 1}{4} \cdot (h x) - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.3

Combine $h$ and $\frac{- 1}{4}$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot - 1}{4} \cdot x - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.4

Combine $\frac{h \cdot - 1}{4}$ and $x$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{1}{8} h^{2} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.5

Combine $h^{2}$ and $\frac{1}{8}$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} - \frac{1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.6

Cancel the common factor of $2$ .

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Step 2.1.2.1.8.6.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- 1}{8} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.6.2

Factor $2$ out of $8$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- 1}{2 (4)} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.6.3

Factor $2$ out of $2 h$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- 1}{2 (4)} \cdot (2 (h)) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.6.4

Cancel the common factor.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- 1}{2 \cdot 4} \cdot (2 h) - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.6.5

Rewrite the expression.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- 1}{4} h - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.7

Combine $\frac{- 1}{4}$ and $h$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} - \frac{1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.8

Cancel the common factor of $2$ .

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Step 2.1.2.1.8.8.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- 1}{8} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.8.2

Factor $2$ out of $8$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- 1}{2 (4)} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.8.3

Factor $2$ out of $2 x$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- 1}{2 (4)} \cdot (2 (x)) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.8.4

Cancel the common factor.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- 1}{2 \cdot 4} \cdot (2 x) - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.8.5

Rewrite the expression.

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- 1}{4} x - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.9

Combine $\frac{- 1}{4}$ and $x$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- x}{4} - \frac{1}{8} \cdot 1 + 8$

Step 2.1.2.1.8.10

Multiply $- 1$ by $1$ .

$f (x + h) = - \frac{x^{2}}{8} + \frac{h \cdot (- 1 x)}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- x}{4} - \frac{1}{8} + 8$

Step 2.1.2.1.9

Simplify each term.

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Step 2.1.2.1.9.1

Factor out negative.

$f (x + h) = - \frac{x^{2}}{8} + \frac{- h x}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- x}{4} - \frac{1}{8} + 8$

Step 2.1.2.1.9.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} + \frac{- h}{4} + \frac{- x}{4} - \frac{1}{8} + 8$

Step 2.1.2.1.9.3

Move the negative in front of the fraction.

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} + \frac{- x}{4} - \frac{1}{8} + 8$

Step 2.1.2.1.9.4

Move the negative in front of the fraction.

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} - \frac{1}{8} + 8$

Step 2.1.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} - \frac{1}{8} + 8 \cdot \frac{8}{8}$

Step 2.1.2.3

Combine $8$ and $\frac{8}{8}$ .

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} - \frac{1}{8} + \frac{8 \cdot 8}{8}$

Step 2.1.2.4

Combine the numerators over the common denominator.

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{- 1 + 8 \cdot 8}{8}$

Step 2.1.2.5

Simplify the numerator.

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Step 2.1.2.5.1

Multiply $8$ by $8$ .

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{- 1 + 64}{8}$

Step 2.1.2.5.2

Add $- 1$ and $64$ .

$f (x + h) = - \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

Step 2.1.2.6

The final answer is $- \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$ .

$- \frac{x^{2}}{8} - \frac{h x}{4} - \frac{h^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

Step 2.2

Reorder.

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Step 2.2.1

Move $- \frac{x^{2}}{8}$ .

$- \frac{h x}{4} - \frac{h^{2}}{8} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

Step 2.2.2

Reorder $- \frac{h x}{4}$ and $- \frac{h^{2}}{8}$ .

$- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

Step 2.3

Find the components of the definition.

$f (x + h) = - \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

$f (x) = - \frac{x^{2}}{8} - \frac{x}{4} + \frac{63}{8}$

$f (x + h) = - \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8}$

$f (x) = - \frac{x^{2}}{8} - \frac{x}{4} + \frac{63}{8}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} - (- \frac{x^{2}}{8} - \frac{x}{4} + \frac{63}{8})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} - - \frac{x^{2}}{8} - - \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- - \frac{x^{2}}{8}$ .

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Step 4.1.2.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + 1 \frac{x^{2}}{8} - - \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.2.1.2

Multiply $\frac{x^{2}}{8}$ by $1$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + \frac{x^{2}}{8} - - \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.2.2

Multiply $- - \frac{x}{4}$ .

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Step 4.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + \frac{x^{2}}{8} + 1 \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.2.2.2

Multiply $\frac{x}{4}$ by $1$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{x^{2}}{8} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + \frac{x^{2}}{8} + \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.3

Add $- \frac{x^{2}}{8}$ and $\frac{x^{2}}{8}$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + 0 + \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.4

Add $- \frac{h^{2}}{8}$ and $0$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} - \frac{x}{4} + \frac{63}{8} + \frac{x}{4} - \frac{63}{8}}{h}$

Step 4.1.5

Add $- \frac{x}{4}$ and $\frac{x}{4}$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + 0 + \frac{63}{8} - \frac{63}{8}}{h}$

Step 4.1.6

Add $- \frac{h^{2}}{8}$ and $0$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + \frac{63}{8} - \frac{63}{8}}{h}$

Step 4.1.7

Combine the numerators over the common denominator.

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + \frac{63 - 63}{8}}{h}$

Step 4.1.8

Subtract $63$ from $63$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + \frac{0}{8}}{h}$

Step 4.1.9

Combine the opposite terms in $- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + \frac{0}{8}$ .

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Step 4.1.9.1

Divide $0$ by $8$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4} + 0}{h}$

Step 4.1.9.2

Add $- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4}$ and $0$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} - \frac{h}{4}}{h}$

Step 4.1.10

To write $- \frac{h x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x}{4} \cdot \frac{2}{2} - \frac{h}{4}}{h}$

Step 4.1.11

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.11.1

Multiply $\frac{h x}{4}$ by $\frac{2}{2}$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x \cdot 2}{4 \cdot 2} - \frac{h}{4}}{h}$

Step 4.1.11.2

Multiply $4$ by $2$ .

$\frac{- \frac{h^{2}}{8} - \frac{h x \cdot 2}{8} - \frac{h}{4}}{h}$

Step 4.1.12

Combine the numerators over the common denominator.

$\frac{\frac{- h^{2} - h x \cdot 2}{8} - \frac{h}{4}}{h}$

Step 4.1.13

Simplify the numerator.

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Step 4.1.13.1

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

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Step 4.1.13.1.1

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

$\frac{\frac{h (- h) - h x \cdot 2}{8} - \frac{h}{4}}{h}$

Step 4.1.13.1.2

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

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

Step 4.1.13.1.3

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

$\frac{\frac{h (- h - 1 x \cdot 2)}{8} - \frac{h}{4}}{h}$

Step 4.1.13.2

Rewrite $- 1 x$ as $- x$ .

$\frac{\frac{h (- h - x \cdot 2)}{8} - \frac{h}{4}}{h}$

Step 4.1.13.3

Multiply $2$ by $- 1$ .

$\frac{\frac{h (- h - 2 x)}{8} - \frac{h}{4}}{h}$

Step 4.1.14

To write $- \frac{h}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{h (- h - 2 x)}{8} - \frac{h}{4} \cdot \frac{2}{2}}{h}$

Step 4.1.15

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.15.1

Multiply $\frac{h}{4}$ by $\frac{2}{2}$ .

$\frac{\frac{h (- h - 2 x)}{8} - \frac{h \cdot 2}{4 \cdot 2}}{h}$

Step 4.1.15.2

Multiply $4$ by $2$ .

$\frac{\frac{h (- h - 2 x)}{8} - \frac{h \cdot 2}{8}}{h}$

Step 4.1.16

Combine the numerators over the common denominator.

$\frac{\frac{h (- h - 2 x) - h \cdot 2}{8}}{h}$

Step 4.1.17

Simplify the numerator.

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Step 4.1.17.1

Factor $h$ out of $h (- h - 2 x) - h \cdot 2$ .

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Step 4.1.17.1.1

Factor $h$ out of $- h \cdot 2$ .

$\frac{\frac{h (- h - 2 x) + h (- 1 \cdot 2)}{8}}{h}$

Step 4.1.17.1.2

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

$\frac{\frac{h (- h - 2 x - 1 \cdot 2)}{8}}{h}$

Step 4.1.17.2

Multiply $- 1$ by $2$ .

$\frac{\frac{h (- h - 2 x - 2)}{8}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (- h - 2 x - 2)}{8} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$\frac{h (- h - 2 x - 2)}{8} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$\frac{- h - 2 x - 2}{8}$

Step 4.4

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

$\frac{- h - (2 x) - 2}{8}$

Step 4.5

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

$\frac{- (h + 2 x) - 2}{8}$

Step 4.6

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (h + 2 x) - 1 (2)}{8}$

Step 4.7

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

$\frac{- (h + 2 x + 2)}{8}$

Step 4.8

Simplify the expression.

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Step 4.8.1

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

$\frac{- 1 (h + 2 x + 2)}{8}$

Step 4.8.2

Move the negative in front of the fraction.

$- \frac{h + 2 x + 2}{8}$

Step 5