Algebra Examples

$\frac{1}{16} x^{2} + \frac{1}{3} x + \frac{4}{9}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{16}$ and $x^{2}$ .

$\frac{x^{2}}{16} + \frac{1}{3} x + \frac{4}{9}$

Step 1.2

Combine $\frac{1}{3}$ and $x$ .

$\frac{x^{2}}{16} + \frac{x}{3} + \frac{4}{9}$

Step 2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{1}{16}$

$b = \frac{1}{3}$

$c = \frac{4}{9}$

Step 3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{3}}{2 (\frac{1}{16})}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} \cdot \frac{1}{2 (\frac{1}{16})}$

Step 4.2.2

Combine $2$ and $\frac{1}{16}$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2}{16}}$

Step 4.2.3

Cancel the common factor of $2$ and $16$ .

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Step 4.2.3.1

Factor $2$ out of $2$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 (1)}{16}}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $2$ out of $16$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 4.2.3.2.2

Cancel the common factor.

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 4.2.3.2.3

Rewrite the expression.

$d = \frac{1}{3} \cdot \frac{1}{\frac{1}{8}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} (1 \cdot 8)$

Step 4.2.5

Multiply $8$ by $1$ .

$d = \frac{1}{3} \cdot 8$

Step 4.2.6

Combine $\frac{1}{3}$ and $8$ .

$d = \frac{8}{3}$

Step 5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{4}{9} - \frac{{(\frac{1}{3})}^{2}}{4 (\frac{1}{16})}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Simplify the numerator.

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Step 5.2.1.1.1

Apply the product rule to $\frac{1}{3}$ .

$e = \frac{4}{9} - \frac{\frac{1^{2}}{3^{2}}}{4 (\frac{1}{16})}$

Step 5.2.1.1.2

One to any power is one.

$e = \frac{4}{9} - \frac{\frac{1}{3^{2}}}{4 (\frac{1}{16})}$

Step 5.2.1.1.3

Raise $3$ to the power of $2$ .

$e = \frac{4}{9} - \frac{\frac{1}{9}}{4 (\frac{1}{16})}$

Step 5.2.1.2

Combine $4$ and $\frac{1}{16}$ .

$e = \frac{4}{9} - \frac{\frac{1}{9}}{\frac{4}{16}}$

Step 5.2.1.3

Cancel the common factor of $4$ and $16$ .

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Step 5.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{4}{9} - \frac{\frac{1}{9}}{\frac{4 (1)}{16}}$

Step 5.2.1.3.2

Cancel the common factors.

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Step 5.2.1.3.2.1

Factor $4$ out of $16$ .

$e = \frac{4}{9} - \frac{\frac{1}{9}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 5.2.1.3.2.2

Cancel the common factor.

$e = \frac{4}{9} - \frac{\frac{1}{9}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 5.2.1.3.2.3

Rewrite the expression.

$e = \frac{4}{9} - \frac{\frac{1}{9}}{\frac{1}{4}}$

Step 5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{4}{9} - (\frac{1}{9} \cdot 4)$

Step 5.2.1.5

Combine $\frac{1}{9}$ and $4$ .

$e = \frac{4}{9} - \frac{4}{9}$

Step 5.2.2

Combine the numerators over the common denominator.

$e = \frac{4 - 4}{9}$

Step 5.2.3

Subtract $4$ from $4$ .

$e = \frac{0}{9}$

Step 5.2.4

Divide $0$ by $9$ .

$e = 0$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{16} {(x + \frac{8}{3})}^{2} + 0$ .

$\frac{1}{16} {(x + \frac{8}{3})}^{2} + 0$