Algebra Examples

$x^{2} + \frac{1}{3} x$

Step 1

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

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

Step 2

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

$a = 1$

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

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.2

Cancel the common factor of $1$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

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

Step 4.2.3

Multiply $\frac{1}{3} \cdot \frac{1}{2}$ .

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Step 4.2.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

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

Step 4.2.3.2

Multiply $3$ by $2$ .

$d = \frac{1}{6}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(\frac{1}{3})}^{2}}{4 \cdot 1}$

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 $\frac{1}{3}$ .

$e = 0 - \frac{\frac{1^{2}}{3^{2}}}{4 \cdot 1}$

Step 5.2.1.1.2

One to any power is one.

$e = 0 - \frac{\frac{1}{3^{2}}}{4 \cdot 1}$

Step 5.2.1.1.3

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

$e = 0 - \frac{\frac{1}{9}}{4 \cdot 1}$

Step 5.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{1}{9}}{4}$

Step 5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{1}{9} \cdot \frac{1}{4})$

Step 5.2.1.4

Multiply $\frac{1}{9} \cdot \frac{1}{4}$ .

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Step 5.2.1.4.1

Multiply $\frac{1}{9}$ by $\frac{1}{4}$ .

$e = 0 - \frac{1}{9 \cdot 4}$

Step 5.2.1.4.2

Multiply $9$ by $4$ .

$e = 0 - \frac{1}{36}$

Step 5.2.2

Subtract $\frac{1}{36}$ from $0$ .

$e = - \frac{1}{36}$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{6})}^{2} - \frac{1}{36}$ .

${(x + \frac{1}{6})}^{2} - \frac{1}{36}$