Precalculus Examples

$y^{2} + 4 x - 2 y + 21 = 0$

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

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

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Step 1.1.1.1

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

$4 x - 2 y + 21 = - y^{2}$

Step 1.1.1.2

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

$4 x + 21 = - y^{2} + 2 y$

Step 1.1.1.3

Subtract $21$ from both sides of the equation.

$4 x = - y^{2} + 2 y - 21$

Step 1.1.2

Divide each term in $4 x = - y^{2} + 2 y - 21$ by $4$ and simplify.

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Step 1.1.2.1

Divide each term in $4 x = - y^{2} + 2 y - 21$ by $4$ .

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{2 y}{4} + \frac{- 21}{4}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{2 y}{4} + \frac{- 21}{4}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{4} + \frac{2 y}{4} + \frac{- 21}{4}$

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

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{2 y}{4} + \frac{- 21}{4}$

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

Factor $2$ out of $2 y$ .

$x = - \frac{y^{2}}{4} + \frac{2 (y)}{4} + \frac{- 21}{4}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $4$ .

$x = - \frac{y^{2}}{4} + \frac{2 y}{2 \cdot 2} + \frac{- 21}{4}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{4} + \frac{2 y}{2 \cdot 2} + \frac{- 21}{4}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{4} + \frac{y}{2} + \frac{- 21}{4}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{y}{2} - \frac{21}{4}$

Step 1.2

Complete the square for $- \frac{y^{2}}{4} + \frac{y}{2} - \frac{21}{4}$ .

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Step 1.2.1

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

$a = - \frac{1}{4}$

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

$c = - \frac{21}{4}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} \cdot \frac{1}{2 (- \frac{1}{4})}$

Step 1.2.3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.2.3.2.2.1

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

$d = \frac{1}{2} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{4})}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

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

Step 1.2.3.2.3

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

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

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.4.2.2

Cancel the common factor.

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

Step 1.2.3.2.4.2.3

Rewrite the expression.

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

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.6

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.1

Factor $2$ out of $- (1 \cdot 2)$ .

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

Step 1.2.3.2.6.2

Cancel the common factor.

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

Step 1.2.3.2.6.3

Rewrite the expression.

$d = - 1 \cdot (1)$

Step 1.2.3.2.7

Multiply $- 1$ by $1$ .

$d = - 1$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - \frac{21}{4} - \frac{{(\frac{1}{2})}^{2}}{4 (- \frac{1}{4})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

$e = - \frac{21}{4} - \frac{\frac{1^{2}}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.2

One to any power is one.

$e = - \frac{21}{4} - \frac{\frac{1}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.3

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

$e = - \frac{21}{4} - \frac{\frac{1}{4}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{21}{4} - \frac{\frac{1}{4}}{- 4 (\frac{1}{4})}$

Step 1.2.4.2.1.2.2

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

$e = - \frac{21}{4} - \frac{\frac{1}{4}}{\frac{- 4}{4}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = - \frac{21}{4} - \frac{\frac{1}{4}}{- 1}$

Step 1.2.4.2.1.4

Move the negative one from the denominator of $\frac{\frac{1}{4}}{- 1}$ .

$e = - \frac{21}{4} - (- 1 \cdot \frac{1}{4})$

Step 1.2.4.2.1.5

Rewrite $- 1 \cdot \frac{1}{4}$ as $- \frac{1}{4}$ .

$e = - \frac{21}{4} - - \frac{1}{4}$

Step 1.2.4.2.1.6

Multiply $- - \frac{1}{4}$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{21}{4} + 1 (\frac{1}{4})$

Step 1.2.4.2.1.6.2

Multiply $\frac{1}{4}$ by $1$ .

$e = - \frac{21}{4} + \frac{1}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 21 + 1}{4}$

Step 1.2.4.2.3

Add $- 21$ and $1$ .

$e = \frac{- 20}{4}$

Step 1.2.4.2.4

Divide $- 20$ by $4$ .

$e = - 5$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(y - 1)}^{2} - 5$ .

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

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{4} \cdot {(y - 1)}^{2} - 5$

Step 2

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

$a = - \frac{1}{4}$

$h = - 5$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 1)$

Step 4