Algebra Examples

$h (x) = \frac{1}{98} x^{2}$

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.

$h (x + h) = \frac{1}{98} \cdot {(x + h)}^{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

$h (x + h) = \frac{1}{98} \cdot ((x + h) (x + h))$

Step 2.1.2.2

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

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Step 2.1.2.2.1

Apply the distributive property.

$h (x + h) = \frac{1}{98} \cdot (x (x + h) + h (x + h))$

Step 2.1.2.2.2

Apply the distributive property.

$h (x + h) = \frac{1}{98} \cdot (x \cdot x + x h + h (x + h))$

Step 2.1.2.2.3

Apply the distributive property.

$h (x + h) = \frac{1}{98} \cdot (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$h (x + h) = \frac{1}{98} \cdot (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

$h (x + h) = \frac{1}{98} \cdot (x^{2} + x h + h x + h^{2})$

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

$h (x + h) = \frac{1}{98} \cdot (x^{2} + h x + h x + h^{2})$

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

$h (x + h) = \frac{1}{98} \cdot (x^{2} + 2 h x + h^{2})$

Step 2.1.2.4

Apply the distributive property.

$h (x + h) = \frac{1}{98} x^{2} + \frac{1}{98} \cdot (2 h x) + \frac{1}{98} h^{2}$

Step 2.1.2.5

Simplify.

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Step 2.1.2.5.1

Combine $\frac{1}{98}$ and $x^{2}$ .

$h (x + h) = \frac{x^{2}}{98} + \frac{1}{98} \cdot (2 h x) + \frac{1}{98} h^{2}$

Step 2.1.2.5.2

Cancel the common factor of $2$ .

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Step 2.1.2.5.2.1

Factor $2$ out of $98$ .

$h (x + h) = \frac{x^{2}}{98} + \frac{1}{2 (49)} \cdot (2 h x) + \frac{1}{98} h^{2}$

Step 2.1.2.5.2.2

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

$h (x + h) = \frac{x^{2}}{98} + \frac{1}{2 (49)} \cdot (2 (h x)) + \frac{1}{98} h^{2}$

Step 2.1.2.5.2.3

Cancel the common factor.

$h (x + h) = \frac{x^{2}}{98} + \frac{1}{2 \cdot 49} \cdot (2 (h x)) + \frac{1}{98} h^{2}$

Step 2.1.2.5.2.4

Rewrite the expression.

$h (x + h) = \frac{x^{2}}{98} + \frac{1}{49} \cdot (h x) + \frac{1}{98} h^{2}$

Step 2.1.2.5.3

Combine $h$ and $\frac{1}{49}$ .

$h (x + h) = \frac{x^{2}}{98} + \frac{h}{49} \cdot x + \frac{1}{98} h^{2}$

Step 2.1.2.5.4

Combine $\frac{h}{49}$ and $x$ .

$h (x + h) = \frac{x^{2}}{98} + \frac{h x}{49} + \frac{1}{98} h^{2}$

Step 2.1.2.5.5

Combine $\frac{1}{98}$ and $h^{2}$ .

$h (x + h) = \frac{x^{2}}{98} + \frac{h x}{49} + \frac{h^{2}}{98}$

Step 2.1.2.6

The final answer is $\frac{x^{2}}{98} + \frac{h x}{49} + \frac{h^{2}}{98}$ .

$\frac{x^{2}}{98} + \frac{h x}{49} + \frac{h^{2}}{98}$

Step 2.2

Reorder.

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Step 2.2.1

Move $\frac{x^{2}}{98}$ .

$\frac{h x}{49} + \frac{h^{2}}{98} + \frac{x^{2}}{98}$

Step 2.2.2

Reorder $\frac{h x}{49}$ and $\frac{h^{2}}{98}$ .

$\frac{h^{2}}{98} + \frac{h x}{49} + \frac{x^{2}}{98}$

Step 2.3

Find the components of the definition.

$h (x + h) = \frac{h^{2}}{98} + \frac{h x}{49} + \frac{x^{2}}{98}$

$h (x) = \frac{x^{2}}{98}$

$h (x + h) = \frac{h^{2}}{98} + \frac{h x}{49} + \frac{x^{2}}{98}$

$h (x) = \frac{x^{2}}{98}$

Step 3

Plug in the components.

$\frac{h (x + h) - h x}{h} = \frac{\frac{h^{2}}{98} + \frac{h x}{49} + \frac{x^{2}}{98} - (\frac{x^{2}}{98})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Subtract $\frac{x^{2}}{98}$ from $\frac{x^{2}}{98}$ .

$\frac{\frac{h^{2}}{98} + \frac{h x}{49} + 0}{h}$

Step 4.1.2

Add $\frac{h^{2}}{98} + \frac{h x}{49}$ and $0$ .

$\frac{\frac{h^{2}}{98} + \frac{h x}{49}}{h}$

Step 4.1.3

To write $\frac{h x}{49}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{h^{2}}{98} + \frac{h x}{49} \cdot \frac{2}{2}}{h}$

Step 4.1.4

Write each expression with a common denominator of $98$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1

Multiply $\frac{h x}{49}$ by $\frac{2}{2}$ .

$\frac{\frac{h^{2}}{98} + \frac{h x \cdot 2}{49 \cdot 2}}{h}$

Step 4.1.4.2

Multiply $49$ by $2$ .

$\frac{\frac{h^{2}}{98} + \frac{h x \cdot 2}{98}}{h}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{\frac{h^{2} + h x \cdot 2}{98}}{h}$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

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

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Step 4.1.6.1.1

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

$\frac{\frac{h \cdot h + h x \cdot 2}{98}}{h}$

Step 4.1.6.1.2

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

$\frac{\frac{h (h) + h (x \cdot 2)}{98}}{h}$

Step 4.1.6.1.3

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

$\frac{\frac{h (h + x \cdot 2)}{98}}{h}$

Step 4.1.6.2

Move $2$ to the left of $x$ .

$\frac{\frac{h (h + 2 x)}{98}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (h + 2 x)}{98} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$\frac{h (h + 2 x)}{98} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$\frac{h + 2 x}{98}$

Step 5