Algebra Examples

$y = \frac{1}{24} x^{2} - \frac{1}{6} x - \frac{35}{6}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{24}$ and $x^{2}$ .

$y = \frac{x^{2}}{24} - \frac{1}{6} x - \frac{35}{6}$

Step 1.1.2

Combine $x$ and $\frac{1}{6}$ .

$y = \frac{x^{2}}{24} - \frac{x}{6} - \frac{35}{6}$

Step 1.2

Complete the square for $\frac{x^{2}}{24} - \frac{x}{6} - \frac{35}{6}$ .

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Step 1.2.1

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

$a = \frac{1}{24}$

$b = - \frac{1}{6}$

$c = - \frac{35}{6}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{- \frac{1}{6}}{2 (\frac{1}{24})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{6} \cdot \frac{1}{2 (\frac{1}{24})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{24}$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2}{24}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $24$ .

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 (1)}{24}}$

Step 1.2.3.2.3.2

Cancel the common factors.

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Step 1.2.3.2.3.2.1

Factor $2$ out of $24$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = - \frac{1}{6} \cdot \frac{1}{\frac{1}{12}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{6} (1 \cdot 12)$

Step 1.2.3.2.5

Multiply $12$ by $1$ .

$d = - \frac{1}{6} \cdot 12$

Step 1.2.3.2.6

Cancel the common factor of $6$ .

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Step 1.2.3.2.6.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$d = \frac{- 1}{6} \cdot 12$

Step 1.2.3.2.6.2

Factor $6$ out of $12$ .

$d = \frac{- 1}{6} \cdot (6 (2))$

Step 1.2.3.2.6.3

Cancel the common factor.

$d = \frac{- 1}{6} \cdot (6 \cdot 2)$

Step 1.2.3.2.6.4

Rewrite the expression.

$d = - 1 \cdot 2$

Step 1.2.3.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - \frac{35}{6} - \frac{{(- \frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $- \frac{1}{6}$ .

$e = - \frac{35}{6} - \frac{{(- 1)}^{2} {(\frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = - \frac{35}{6} - \frac{1 {(\frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{6}$ .

$e = - \frac{35}{6} - \frac{1 \frac{1^{2}}{6^{2}}}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.1.4

One to any power is one.

$e = - \frac{35}{6} - \frac{1 \frac{1}{6^{2}}}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.1.5

Raise $6$ to the power of $2$ .

$e = - \frac{35}{6} - \frac{1 (\frac{1}{36})}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{1}{36}$ by $1$ .

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{4 (\frac{1}{24})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{24}$ .

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{\frac{4}{24}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $24$ .

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{\frac{4 (1)}{24}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $24$ .

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = - \frac{35}{6} - \frac{\frac{1}{36}}{\frac{1}{6}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{35}{6} - (\frac{1}{36} \cdot 6)$

Step 1.2.4.2.1.5

Cancel the common factor of $6$ .

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Step 1.2.4.2.1.5.1

Factor $6$ out of $36$ .

$e = - \frac{35}{6} - (\frac{1}{6 (6)} \cdot 6)$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = - \frac{35}{6} - (\frac{1}{6 \cdot 6} \cdot 6)$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = - \frac{35}{6} - \frac{1}{6}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 35 - 1}{6}$

Step 1.2.4.2.3

Subtract $1$ from $- 35$ .

$e = \frac{- 36}{6}$

Step 1.2.4.2.4

Divide $- 36$ by $6$ .

$e = - 6$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{24} {(x - 2)}^{2} - 6$ .

$\frac{1}{24} {(x - 2)}^{2} - 6$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{24} \cdot {(x - 2)}^{2} - 6$

Step 2

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

$a = \frac{1}{24}$

$h = 2$

$k = - 6$

Step 3

Find the vertex $(h, k)$ .

$(2, - 6)$

Step 4

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

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Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{24}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{1}{24}$ .

$\frac{1}{\frac{4}{24}}$

Step 4.3.2

Cancel the common factor of $4$ and $24$ .

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Step 4.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{24}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $4$ out of $24$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{6}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 6$

Step 4.3.4

Multiply $6$ by $1$ .

$6$

Step 5

Find the focus.

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Step 5.1

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 5.2

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

$(2, 0)$

Step 6