Algebra Examples

$g (x) = \frac{2}{3} x (x + 8)$

Step 1

Write $g (x) = \frac{2}{3} x (x + 8)$ as an equation.

$y = \frac{2}{3} x (x + 8)$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Isolate $y$ to the left side of the equation.

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Step 2.1.1

Remove parentheses.

$y = \frac{2}{3} \cdot (x (x + 8))$

Step 2.1.2

Remove parentheses.

$y = \frac{2}{3} \cdot (x (x + 8))$

Step 2.1.3

Simplify $\frac{2}{3} \cdot (x (x + 8))$ .

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Step 2.1.3.1

Apply the distributive property.

$y = \frac{2}{3} \cdot (x \cdot x + x \cdot 8)$

Step 2.1.3.2

Simplify the expression.

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Step 2.1.3.2.1

Multiply $x$ by $x$ .

$y = \frac{2}{3} \cdot (x^{2} + x \cdot 8)$

Step 2.1.3.2.2

Move $8$ to the left of $x$ .

$y = \frac{2}{3} \cdot (x^{2} + 8 \cdot x)$

Step 2.1.3.3

Apply the distributive property.

$y = \frac{2}{3} x^{2} + \frac{2}{3} (8 x)$

Step 2.1.3.4

Combine $\frac{2}{3}$ and $x^{2}$ .

$y = \frac{2 x^{2}}{3} + \frac{2}{3} (8 x)$

Step 2.1.3.5

Multiply $\frac{2}{3} (8 x)$ .

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Step 2.1.3.5.1

Combine $8$ and $\frac{2}{3}$ .

$y = \frac{2 x^{2}}{3} + \frac{8 \cdot 2}{3} x$

Step 2.1.3.5.2

Multiply $8$ by $2$ .

$y = \frac{2 x^{2}}{3} + \frac{16}{3} x$

Step 2.1.3.5.3

Combine $\frac{16}{3}$ and $x$ .

$y = \frac{2 x^{2}}{3} + \frac{16 x}{3}$

Step 2.2

Complete the square for $\frac{2 x^{2}}{3} + \frac{16 x}{3}$ .

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Step 2.2.1

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

$a = \frac{2}{3}$

$b = \frac{16}{3}$

$c = 0$

Step 2.2.2

Consider the vertex form of a parabola.

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $16$ .

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

Step 2.2.3.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.3

Multiply $\frac{8}{3}$ by $\frac{1}{\frac{2}{3}}$ .

$d = \frac{8}{3 (\frac{2}{3})}$

Step 2.2.3.2.4

Combine $3$ and $\frac{2}{3}$ .

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

Step 2.2.3.2.5

Multiply $3$ by $2$ .

$d = \frac{8}{\frac{6}{3}}$

Step 2.2.3.2.6

Cancel the common factor of $6$ and $3$ .

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Step 2.2.3.2.6.1

Factor $3$ out of $6$ .

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

Step 2.2.3.2.6.2

Cancel the common factors.

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Step 2.2.3.2.6.2.1

Factor $3$ out of $3$ .

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

Step 2.2.3.2.6.2.2

Cancel the common factor.

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

Step 2.2.3.2.6.2.3

Rewrite the expression.

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

Step 2.2.3.2.6.2.4

Divide $2$ by $1$ .

$d = \frac{8}{2}$

Step 2.2.3.2.7

Cancel the common factor of $8$ and $2$ .

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Step 2.2.3.2.7.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2}$

Step 2.2.3.2.7.2

Cancel the common factors.

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Step 2.2.3.2.7.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 4}{2 (1)}$

Step 2.2.3.2.7.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 1}$

Step 2.2.3.2.7.2.3

Rewrite the expression.

$d = \frac{4}{1}$

Step 2.2.3.2.7.2.4

Divide $4$ by $1$ .

$d = 4$

Step 2.2.4

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

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Step 2.2.4.1

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

$e = 0 - \frac{{(\frac{16}{3})}^{2}}{4 (\frac{2}{3})}$

Step 2.2.4.2

Simplify the right side.

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Step 2.2.4.2.1

Simplify each term.

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Step 2.2.4.2.1.1

Simplify the numerator.

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Step 2.2.4.2.1.1.1

Apply the product rule to $\frac{16}{3}$ .

$e = 0 - \frac{\frac{16^{2}}{3^{2}}}{4 (\frac{2}{3})}$

Step 2.2.4.2.1.1.2

Raise $16$ to the power of $2$ .

$e = 0 - \frac{\frac{256}{3^{2}}}{4 (\frac{2}{3})}$

Step 2.2.4.2.1.1.3

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

$e = 0 - \frac{\frac{256}{9}}{4 (\frac{2}{3})}$

Step 2.2.4.2.1.2

Combine $4$ and $\frac{2}{3}$ .

$e = 0 - \frac{\frac{256}{9}}{\frac{4 \cdot 2}{3}}$

Step 2.2.4.2.1.3

Multiply $4$ by $2$ .

$e = 0 - \frac{\frac{256}{9}}{\frac{8}{3}}$

Step 2.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{256}{9} \cdot \frac{3}{8})$

Step 2.2.4.2.1.5

Cancel the common factor of $8$ .

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Step 2.2.4.2.1.5.1

Factor $8$ out of $256$ .

$e = 0 - (\frac{8 (32)}{9} \cdot \frac{3}{8})$

Step 2.2.4.2.1.5.2

Cancel the common factor.

$e = 0 - (\frac{8 \cdot 32}{9} \cdot \frac{3}{8})$

Step 2.2.4.2.1.5.3

Rewrite the expression.

$e = 0 - (\frac{32}{9} \cdot 3)$

Step 2.2.4.2.1.6

Cancel the common factor of $3$ .

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Step 2.2.4.2.1.6.1

Factor $3$ out of $9$ .

$e = 0 - (\frac{32}{3 (3)} \cdot 3)$

Step 2.2.4.2.1.6.2

Cancel the common factor.

$e = 0 - (\frac{32}{3 \cdot 3} \cdot 3)$

Step 2.2.4.2.1.6.3

Rewrite the expression.

$e = 0 - \frac{32}{3}$

Step 2.2.4.2.2

Subtract $\frac{32}{3}$ from $0$ .

$e = - \frac{32}{3}$

Step 2.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{2}{3} {(x + 4)}^{2} - \frac{32}{3}$ .

$\frac{2}{3} {(x + 4)}^{2} - \frac{32}{3}$

Step 2.3

Set $y$ equal to the new right side.

$y = \frac{2}{3} \cdot {(x + 4)}^{2} - \frac{32}{3}$

Step 3

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{2}{3}$

$h = - 4$

$k = - \frac{32}{3}$

Step 4

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(- 4, - \frac{32}{3})$

Step 6

Find $p$ , the distance from the vertex to the focus.

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Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 6.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{2}{3}}$

Step 6.3

Simplify.

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Step 6.3.1

Combine $4$ and $\frac{2}{3}$ .

$\frac{1}{\frac{4 \cdot 2}{3}}$

Step 6.3.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{8}{3}}$

Step 6.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{8})$

Step 6.3.4

Multiply $\frac{3}{8}$ by $1$ .

$\frac{3}{8}$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 7.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 4, - \frac{247}{24})$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - 4$

Step 9