Precalculus Examples

$2 x - y^{2} = 4 y + 10$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Add $y^{2}$ to both sides of the equation.

$2 x = 4 y + 10 + y^{2}$

Step 1.1.2

Divide each term in $2 x = 4 y + 10 + y^{2}$ by $2$ and simplify.

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Step 1.1.2.1

Divide each term in $2 x = 4 y + 10 + y^{2}$ by $2$ .

$\frac{2 x}{2} = \frac{4 y}{2} + \frac{10}{2} + \frac{y^{2}}{2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4 y}{2} + \frac{10}{2} + \frac{y^{2}}{2}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 y}{2} + \frac{10}{2} + \frac{y^{2}}{2}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

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Step 1.1.2.3.1.1.1

Factor $2$ out of $4 y$ .

$x = \frac{2 (2 y)}{2} + \frac{10}{2} + \frac{y^{2}}{2}$

Step 1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.2.3.1.1.2.1

Factor $2$ out of $2$ .

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

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.1.2.4

Divide $2 y$ by $1$ .

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

Step 1.1.2.3.1.2

Divide $10$ by $2$ .

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

Step 1.2

Complete the square for $2 y + 5 + \frac{y^{2}}{2}$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Move $5$ .

$2 y + \frac{y^{2}}{2} + 5$

Step 1.2.1.2

Reorder $2 y$ and $\frac{y^{2}}{2}$ .

$\frac{y^{2}}{2} + 2 y + 5$

Step 1.2.2

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

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

$b = 2$

$c = 5$

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.1.2

Rewrite the expression.

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

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Step 1.2.4.2.3

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

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

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 $2$ to the power of $2$ .

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

Step 1.2.5.2.1.2

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

$e = 5 - \frac{4}{\frac{4}{2}}$

Step 1.2.5.2.1.3

Divide $4$ by $2$ .

$e = 5 - \frac{4}{2}$

Step 1.2.5.2.1.4

Divide $4$ by $2$ .

$e = 5 - 1 \cdot 2$

Step 1.2.5.2.1.5

Multiply $- 1$ by $2$ .

$e = 5 - 2$

Step 1.2.5.2.2

Subtract $2$ from $5$ .

$e = 3$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(y + 2)}^{2} + 3$ .

$\frac{1}{2} {(y + 2)}^{2} + 3$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{2} \cdot {(y + 2)}^{2} + 3$

Step 2

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

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

$h = 3$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(3, - 2)$

Step 4