Precalculus Examples

$y^{2} - 2 x - 2 y - 9 = 0$

Step 1

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

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Step 1.1

Add $2 x$ to both sides of the equation.

$y^{2} - 2 y - 9 = 2 x$

Step 1.2

Add $9$ to both sides of the equation.

$y^{2} - 2 y = 2 x + 9$

Step 2

Complete the square.

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Step 2.1

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

$a = 1$

$b = - 2$

$c = 0 = 2 x + 9$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = 2 x + 9$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 2.4

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

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Step 2.4.1

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

$e = 0 - \frac{{(- 2)}^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

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

$e = 0 - \frac{4}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Cancel the common factor of $4$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

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

Step 2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 2.5

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

${(y - 1)}^{2} - 1 = 2 x + 9$

Step 3

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

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Step 3.1

Add $1$ to both sides of the equation.

${(y - 1)}^{2} = 2 x + 9 + 1$

Step 3.2

Add $9$ and $1$ .

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

Step 4

Factor $2$ out of $2 x + 10$ .

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Step 4.1

Factor $2$ out of $2 x$ .

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

Step 4.2

Factor $2$ out of $10$ .

${(y - 1)}^{2} = 2 x + 2 \cdot 5$

Step 4.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

${(y - 1)}^{2} = 2 (x + 5)$