Algebra Examples

$y = \frac{1}{10} x^{2} - \frac{1}{5} x + \frac{18}{5}$

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 $\frac{1}{10}$ and $x^{2}$ .

$y = \frac{x^{2}}{10} - \frac{1}{5} x + \frac{18}{5}$

Step 1.1.2

Combine $x$ and $\frac{1}{5}$ .

$y = \frac{x^{2}}{10} - \frac{x}{5} + \frac{18}{5}$

Step 1.2

Complete the square for $\frac{x^{2}}{10} - \frac{x}{5} + \frac{18}{5}$ .

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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 $c$ .

$a = \frac{1}{10}$

$b = - \frac{1}{5}$

$c = \frac{18}{5}$

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

$d = \frac{- \frac{1}{5}}{2 (\frac{1}{10})}$

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.

$d = - \frac{1}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{10}$ .

$d = - \frac{1}{5} \cdot \frac{1}{\frac{2}{10}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $10$ .

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{1}{5} \cdot \frac{1}{\frac{2 (1)}{10}}$

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 $10$ .

$d = - \frac{1}{5} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 5}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = - \frac{1}{5} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 5}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = - \frac{1}{5} \cdot \frac{1}{\frac{1}{5}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{5} (1 \cdot 5)$

Step 1.2.3.2.5

Cancel the common factor of $5$ .

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Step 1.2.3.2.5.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$d = \frac{- 1}{5} (1 \cdot 5)$

Step 1.2.3.2.5.2

Factor $5$ out of $1 \cdot 5$ .

$d = \frac{- 1}{5} (5 \cdot 1)$

Step 1.2.3.2.5.3

Cancel the common factor.

$d = \frac{- 1}{5} (5 \cdot 1)$

Step 1.2.3.2.5.4

Rewrite the expression.

$d = - 1$

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

$e = \frac{18}{5} - \frac{{(- \frac{1}{5})}^{2}}{4 (\frac{1}{10})}$

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 $- \frac{1}{5}$ .

$e = \frac{18}{5} - \frac{{(- 1)}^{2} {(\frac{1}{5})}^{2}}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.1.2

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

$e = \frac{18}{5} - \frac{1 {(\frac{1}{5})}^{2}}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{5}$ .

$e = \frac{18}{5} - \frac{1 \frac{1^{2}}{5^{2}}}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.1.4

One to any power is one.

$e = \frac{18}{5} - \frac{1 \frac{1}{5^{2}}}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.1.5

Raise $5$ to the power of $2$ .

$e = \frac{18}{5} - \frac{1 (\frac{1}{25})}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{1}{25}$ by $1$ .

$e = \frac{18}{5} - \frac{\frac{1}{25}}{4 (\frac{1}{10})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{10}$ .

$e = \frac{18}{5} - \frac{\frac{1}{25}}{\frac{4}{10}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $10$ .

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Step 1.2.4.2.1.3.1

Factor $2$ out of $4$ .

$e = \frac{18}{5} - \frac{\frac{1}{25}}{\frac{2 (2)}{10}}$

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 $2$ out of $10$ .

$e = \frac{18}{5} - \frac{\frac{1}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{18}{5} - \frac{\frac{1}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{18}{5} - \frac{\frac{1}{25}}{\frac{2}{5}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{18}{5} - (\frac{1}{25} \cdot \frac{5}{2})$

Step 1.2.4.2.1.5

Cancel the common factor of $5$ .

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Step 1.2.4.2.1.5.1

Factor $5$ out of $25$ .

$e = \frac{18}{5} - (\frac{1}{5 (5)} \cdot \frac{5}{2})$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = \frac{18}{5} - (\frac{1}{5 \cdot 5} \cdot \frac{5}{2})$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = \frac{18}{5} - (\frac{1}{5} \cdot \frac{1}{2})$

Step 1.2.4.2.1.6

Multiply $\frac{1}{5}$ by $\frac{1}{2}$ .

$e = \frac{18}{5} - \frac{1}{5 \cdot 2}$

Step 1.2.4.2.1.7

Multiply $5$ by $2$ .

$e = \frac{18}{5} - \frac{1}{10}$

Step 1.2.4.2.2

To write $\frac{18}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = \frac{18}{5} \cdot \frac{2}{2} - \frac{1}{10}$

Step 1.2.4.2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.2.3.1

Multiply $\frac{18}{5}$ by $\frac{2}{2}$ .

$e = \frac{18 \cdot 2}{5 \cdot 2} - \frac{1}{10}$

Step 1.2.4.2.3.2

Multiply $5$ by $2$ .

$e = \frac{18 \cdot 2}{10} - \frac{1}{10}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{18 \cdot 2 - 1}{10}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $18$ by $2$ .

$e = \frac{36 - 1}{10}$

Step 1.2.4.2.5.2

Subtract $1$ from $36$ .

$e = \frac{35}{10}$

Step 1.2.4.2.6

Cancel the common factor of $35$ and $10$ .

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Step 1.2.4.2.6.1

Factor $5$ out of $35$ .

$e = \frac{5 (7)}{10}$

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $5$ out of $10$ .

$e = \frac{5 \cdot 7}{5 \cdot 2}$

Step 1.2.4.2.6.2.2

Cancel the common factor.

$e = \frac{5 \cdot 7}{5 \cdot 2}$

Step 1.2.4.2.6.2.3

Rewrite the expression.

$e = \frac{7}{2}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{10} {(x - 1)}^{2} + \frac{7}{2}$ .

$\frac{1}{10} {(x - 1)}^{2} + \frac{7}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{10} \cdot {(x - 1)}^{2} + \frac{7}{2}$

Step 2

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

$a = \frac{1}{10}$

$h = 1$

$k = \frac{7}{2}$

Step 3

Find the vertex $(h, k)$ .

$(1, \frac{7}{2})$

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.

$\frac{1}{4 \cdot \frac{1}{10}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{1}{10}$ .

$\frac{1}{\frac{4}{10}}$

Step 4.3.2

Cancel the common factor of $4$ and $10$ .

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Step 4.3.2.1

Factor $2$ out of $4$ .

$\frac{1}{\frac{2 (2)}{10}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $2$ out of $10$ .

$\frac{1}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{2 \cdot 2}{2 \cdot 5}}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{2}{5}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{5}{2})$

Step 4.3.4

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2}$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 5.2

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

$y = 1$

Step 6