Algebra Examples

$y = - \frac{1}{4} x^{2} - \frac{1}{2} x - \frac{1}{4}$

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 $x^{2}$ and $\frac{1}{4}$ .

$y = - \frac{x^{2}}{4} - \frac{1}{2} x - \frac{1}{4}$

Step 1.1.2

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

$y = - \frac{x^{2}}{4} - \frac{x}{2} - \frac{1}{4}$

Step 1.2

Complete the square for $- \frac{x^{2}}{4} - \frac{x}{2} - \frac{1}{4}$ .

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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 $c$ .

$a = - \frac{1}{4}$

$b = - \frac{1}{2}$

$c = - \frac{1}{4}$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{4}$ .

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

Step 1.2.3.2.4

Cancel the common factor of $2$ and $4$ .

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.4.2.2

Cancel the common factor.

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

Step 1.2.3.2.4.2.3

Rewrite the expression.

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

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.6

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.1

Factor $2$ out of $1 \cdot 2$ .

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

Step 1.2.3.2.6.2

Cancel the common factor.

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

Step 1.2.3.2.6.3

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

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

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 $- \frac{1}{2}$ .

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

Step 1.2.4.2.1.1.2

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

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

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2.4.2.1.1.4

One to any power is one.

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

Step 1.2.4.2.1.1.5

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

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

Step 1.2.4.2.1.1.6

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

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

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 1.2.4.2.1.2.2

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

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

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

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

Step 1.2.4.2.1.4

Move the negative one from the denominator of $\frac{\frac{1}{4}}{- 1}$ .

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

Step 1.2.4.2.1.5

Rewrite $- 1 \cdot \frac{1}{4}$ as $- \frac{1}{4}$ .

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

Step 1.2.4.2.1.6

Multiply $- - \frac{1}{4}$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4.2.1.6.2

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

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

Step 1.2.4.2.2

Combine the numerators over the common denominator.

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

Step 1.2.4.2.3

Add $- 1$ and $1$ .

$e = \frac{0}{4}$

Step 1.2.4.2.4

Divide $0$ by $4$ .

$e = 0$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 1)}^{2} + 0$ .

$- \frac{1}{4} {(x + 1)}^{2} + 0$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 1)}^{2} + 0$

Step 2

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

$a = - \frac{1}{4}$

$h = - 1$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 0)$

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.

$\frac{1}{4 \cdot (- \frac{1}{4})}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{4})}$

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

$- \frac{1}{\frac{4}{4}}$

Step 4.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 4.3.4

Cancel the common factor of $1$ .

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Step 4.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 4.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 4.3.5

Multiply $- 1$ by $1$ .

$- 1$

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.

$(- 1, - 1)$

Step 6