Algebra Examples

$x + 3 y^{2} + 12 y = 18$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1

Subtract $3 y^{2}$ from both sides of the equation.

$x + 12 y = 18 - 3 y^{2}$

Step 1.1.2

Subtract $12 y$ from both sides of the equation.

$x = 18 - 3 y^{2} - 12 y$

Step 1.2

Complete the square for $18 - 3 y^{2} - 12 y$ .

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Step 1.2.1

Move $18$ .

$- 3 y^{2} - 12 y + 18$

Step 1.2.2

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

$a = - 3$

$b = - 12$

$c = 18$

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

$d = \frac{- 12}{2 \cdot - 3}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot - 3}$

Step 1.2.4.2.1.2

Cancel the common factors.

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Step 1.2.4.2.1.2.1

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 \cdot - 6}{2 (- 3)}$

Step 1.2.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot - 3}$

Step 1.2.4.2.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.2

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

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Step 1.2.4.2.2.1

Factor $- 3$ out of $- 6$ .

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

Step 1.2.4.2.2.2

Cancel the common factors.

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Step 1.2.4.2.2.2.1

Factor $- 3$ out of $- 3$ .

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

Step 1.2.4.2.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.2.2.3

Rewrite the expression.

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

Step 1.2.4.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

$e = 18 - \frac{{(- 12)}^{2}}{4 \cdot - 3}$

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

Raise $- 12$ to the power of $2$ .

$e = 18 - \frac{144}{4 \cdot - 3}$

Step 1.2.5.2.1.2

Multiply $4$ by $- 3$ .

$e = 18 - \frac{144}{- 12}$

Step 1.2.5.2.1.3

Divide $144$ by $- 12$ .

$e = 18 - - 12$

Step 1.2.5.2.1.4

Multiply $- 1$ by $- 12$ .

$e = 18 + 12$

Step 1.2.5.2.2

Add $18$ and $12$ .

$e = 30$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(y + 2)}^{2} + 30$ .

$- 3 {(y + 2)}^{2} + 30$

Step 1.3

Set $x$ equal to the new right side.

$x = - 3 {(y + 2)}^{2} + 30$

Step 2

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

$a = - 3$

$h = 30$

$k = - 2$

Step 3

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(30, - 2)$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

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

$(\frac{359}{12}, - 2)$

Step 7

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

$y = - 2$

Step 8