Precalculus Examples

$y^{2} + 8 x - 2 y - 15 = 0$

Step 1

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

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Step 1.1

Subtract $8 x$ from both sides of the equation.

$y^{2} - 2 y - 15 = - 8 x$

Step 1.2

Add $15$ to both sides of the equation.

$y^{2} - 2 y = - 8 x + 15$

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 $c$ .

$a = 1$

$b = - 2$

$c = 0 = - 8 x + 15$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = - 8 x + 15$

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 ${(y - 1)}^{2} - 1$ .

${(y - 1)}^{2} - 1 = - 8 x + 15$

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.

${(y - 1)}^{2} = - 8 x + 15 + 1$

Step 3.2

Add $15$ and $1$ .

${(y - 1)}^{2} = - 8 x + 16$

Step 4

Factor.

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Step 4.1

Factor $8$ out of $- 8 x + 16$ .

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Step 4.1.1

Factor $8$ out of $- 8 x$ .

${(y - 1)}^{2} = 8 (- x) + 16$

Step 4.1.2

Factor $8$ out of $16$ .

${(y - 1)}^{2} = 8 (- x) + 8 (2)$

Step 4.1.3

Factor $8$ out of $8 (- x) + 8 (2)$ .

${(y - 1)}^{2} = 8 (- x + 2)$

Step 4.2

Factor $- 1$ out of $- x$ .

${(y - 1)}^{2} = 8 (- (x) + 2)$

Step 4.3

Rewrite $2$ as $- 1 (- 2)$ .

${(y - 1)}^{2} = 8 (- (x) - 1 (- 2))$

Step 4.4

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

${(y - 1)}^{2} = 8 (- (x - 2))$

Step 4.5

Multiply $- 1$ by $8$ .

${(y - 1)}^{2} = - 8 (x - 2)$