Algebra Examples

$- 12 (y - 2) = {(x + 1)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Divide each term in $- 12 (y - 2) = {(x + 1)}^{2}$ by $- 12$ and simplify.

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Step 1.1.1.1

Divide each term in $- 12 (y - 2) = {(x + 1)}^{2}$ by $- 12$ .

$\frac{- 12 (y - 2)}{- 12} = \frac{{(x + 1)}^{2}}{- 12}$

Step 1.1.1.2

Simplify the left side.

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Step 1.1.1.2.1

Cancel the common factor of $- 12$ .

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Step 1.1.1.2.1.1

Cancel the common factor.

$\frac{- 12 (y - 2)}{- 12} = \frac{{(x + 1)}^{2}}{- 12}$

Step 1.1.1.2.1.2

Divide $y - 2$ by $1$ .

$y - 2 = \frac{{(x + 1)}^{2}}{- 12}$

Step 1.1.1.3

Simplify the right side.

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Step 1.1.1.3.1

Move the negative in front of the fraction.

$y - 2 = - \frac{{(x + 1)}^{2}}{12}$

Step 1.1.2

Add $2$ to both sides of the equation.

$y = - \frac{{(x + 1)}^{2}}{12} + 2$

Step 1.1.3

Reorder terms.

$y = - \frac{1}{12} \cdot {(x + 1)}^{2} + 2$

Step 1.2

Complete the square for $- \frac{1}{12} \cdot {(x + 1)}^{2} + 2$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$- \frac{1}{12} \cdot ((x + 1) (x + 1)) + 2$

Step 1.2.1.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$- \frac{1}{12} \cdot (x (x + 1) + 1 (x + 1)) + 2$

Step 1.2.1.1.2.2

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot 1 + 1 (x + 1)) + 2$

Step 1.2.1.1.2.3

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 2$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$- \frac{1}{12} \cdot (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 2$

Step 1.2.1.1.3.1.2

Multiply $x$ by $1$ .

$- \frac{1}{12} \cdot (x^{2} + x + 1 x + 1 \cdot 1) + 2$

Step 1.2.1.1.3.1.3

Multiply $x$ by $1$ .

$- \frac{1}{12} \cdot (x^{2} + x + x + 1 \cdot 1) + 2$

Step 1.2.1.1.3.1.4

Multiply $1$ by $1$ .

$- \frac{1}{12} \cdot (x^{2} + x + x + 1) + 2$

Step 1.2.1.1.3.2

Add $x$ and $x$ .

$- \frac{1}{12} \cdot (x^{2} + 2 x + 1) + 2$

Step 1.2.1.1.4

Apply the distributive property.

$- \frac{1}{12} x^{2} - \frac{1}{12} (2 x) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $x^{2}$ and $\frac{1}{12}$ .

$- \frac{x^{2}}{12} - \frac{1}{12} (2 x) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.2.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$- \frac{x^{2}}{12} + \frac{- 1}{12} (2 x) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.2.2

Factor $2$ out of $12$ .

$- \frac{x^{2}}{12} + \frac{- 1}{2 (6)} (2 x) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.2.3

Factor $2$ out of $2 x$ .

$- \frac{x^{2}}{12} + \frac{- 1}{2 (6)} (2 (x)) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.2.4

Cancel the common factor.

$- \frac{x^{2}}{12} + \frac{- 1}{2 \cdot 6} (2 x) - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.2.5

Rewrite the expression.

$- \frac{x^{2}}{12} + \frac{- 1}{6} x - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.3

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

$- \frac{x^{2}}{12} + \frac{- x}{6} - \frac{1}{12} \cdot 1 + 2$

Step 1.2.1.1.5.4

Multiply $- 1$ by $1$ .

$- \frac{x^{2}}{12} + \frac{- x}{6} - \frac{1}{12} + 2$

Step 1.2.1.1.6

Move the negative in front of the fraction.

$- \frac{x^{2}}{12} - \frac{x}{6} - \frac{1}{12} + 2$

Step 1.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$- \frac{x^{2}}{12} - \frac{x}{6} - \frac{1}{12} + 2 \cdot \frac{12}{12}$

Step 1.2.1.3

Combine $2$ and $\frac{12}{12}$ .

$- \frac{x^{2}}{12} - \frac{x}{6} - \frac{1}{12} + \frac{2 \cdot 12}{12}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$- \frac{x^{2}}{12} - \frac{x}{6} + \frac{- 1 + 2 \cdot 12}{12}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $2$ by $12$ .

$- \frac{x^{2}}{12} - \frac{x}{6} + \frac{- 1 + 24}{12}$

Step 1.2.1.5.2

Add $- 1$ and $24$ .

$- \frac{x^{2}}{12} - \frac{x}{6} + \frac{23}{12}$

Step 1.2.2

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

$a = - \frac{1}{12}$

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

$c = \frac{23}{12}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Dividing two negative values results in a positive value.

$d = \frac{\frac{1}{6}}{2 (\frac{1}{12})}$

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.3

Combine $2$ and $\frac{1}{12}$ .

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

Step 1.2.4.2.4

Cancel the common factor of $2$ and $12$ .

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Step 1.2.4.2.4.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.4.2

Cancel the common factors.

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Step 1.2.4.2.4.2.1

Factor $2$ out of $12$ .

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

Step 1.2.4.2.4.2.2

Cancel the common factor.

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

Step 1.2.4.2.4.2.3

Rewrite the expression.

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

Step 1.2.4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.6

Cancel the common factor of $6$ .

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Step 1.2.4.2.6.1

Factor $6$ out of $1 \cdot 6$ .

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

Step 1.2.4.2.6.2

Cancel the common factor.

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

Step 1.2.4.2.6.3

Rewrite the expression.

$d = 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = \frac{23}{12} - \frac{{(- \frac{1}{6})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

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

$e = \frac{23}{12} - \frac{{(- 1)}^{2} {(\frac{1}{6})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.2

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

$e = \frac{23}{12} - \frac{1 {(\frac{1}{6})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.3

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

$e = \frac{23}{12} - \frac{1 \frac{1^{2}}{6^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.4

One to any power is one.

$e = \frac{23}{12} - \frac{1 \frac{1}{6^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.5

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

$e = \frac{23}{12} - \frac{1 (\frac{1}{36})}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.6

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

$e = \frac{23}{12} - \frac{\frac{1}{36}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.2

Simplify the denominator.

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{23}{12} - \frac{\frac{1}{36}}{- 4 (\frac{1}{12})}$

Step 1.2.5.2.1.2.2

Combine $- 4$ and $\frac{1}{12}$ .

$e = \frac{23}{12} - \frac{\frac{1}{36}}{\frac{- 4}{12}}$

Step 1.2.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.5.2.1.3.1

Cancel the common factor of $- 4$ and $12$ .

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Step 1.2.5.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = \frac{23}{12} - \frac{\frac{1}{36}}{\frac{4 (- 1)}{12}}$

Step 1.2.5.2.1.3.1.2

Cancel the common factors.

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Step 1.2.5.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = \frac{23}{12} - \frac{\frac{1}{36}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.2

Cancel the common factor.

$e = \frac{23}{12} - \frac{\frac{1}{36}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.3

Rewrite the expression.

$e = \frac{23}{12} - \frac{\frac{1}{36}}{\frac{- 1}{3}}$

Step 1.2.5.2.1.3.2

Move the negative in front of the fraction.

$e = \frac{23}{12} - \frac{\frac{1}{36}}{- \frac{1}{3}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{23}{12} - (\frac{1}{36} (- 1 \cdot 3))$

Step 1.2.5.2.1.5

Cancel the common factor of $3$ .

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Step 1.2.5.2.1.5.1

Factor $3$ out of $36$ .

$e = \frac{23}{12} - (\frac{1}{3 (12)} (- 1 \cdot 3))$

Step 1.2.5.2.1.5.2

Factor $3$ out of $- 1 \cdot 3$ .

$e = \frac{23}{12} - (\frac{1}{3 (12)} (3 \cdot - 1))$

Step 1.2.5.2.1.5.3

Cancel the common factor.

$e = \frac{23}{12} - (\frac{1}{3 \cdot 12} (3 \cdot - 1))$

Step 1.2.5.2.1.5.4

Rewrite the expression.

$e = \frac{23}{12} - (\frac{1}{12} \cdot - 1)$

Step 1.2.5.2.1.6

Combine $\frac{1}{12}$ and $- 1$ .

$e = \frac{23}{12} - \frac{- 1}{12}$

Step 1.2.5.2.1.7

Move the negative in front of the fraction.

$e = \frac{23}{12} - - \frac{1}{12}$

Step 1.2.5.2.1.8

Multiply $- - \frac{1}{12}$ .

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Step 1.2.5.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = \frac{23}{12} + 1 (\frac{1}{12})$

Step 1.2.5.2.1.8.2

Multiply $\frac{1}{12}$ by $1$ .

$e = \frac{23}{12} + \frac{1}{12}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{23 + 1}{12}$

Step 1.2.5.2.3

Add $23$ and $1$ .

$e = \frac{24}{12}$

Step 1.2.5.2.4

Divide $24$ by $12$ .

$e = 2$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(x + 1)}^{2} + 2$ .

$- \frac{1}{12} {(x + 1)}^{2} + 2$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{12} \cdot {(x + 1)}^{2} + 2$

Step 2

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

$a = - \frac{1}{12}$

$h = - 1$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 2)$

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.

$\frac{1}{4 \cdot (- \frac{1}{12})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{12})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{12})}$

Step 4.3.2

Combine $4$ and $\frac{1}{12}$ .

$- \frac{1}{\frac{4}{12}}$

Step 4.3.3

Cancel the common factor of $4$ and $12$ .

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Step 4.3.3.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{12}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $4$ out of $12$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{1}{3}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 4.3.5

Multiply $- (1 \cdot 3)$ .

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Step 4.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 4.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

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.

$y = 5$

Step 6