Precalculus Examples

$y^{2} + 4 y - x + 1 = 0$

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.

$4 y - x + 1 = - y^{2}$

Step 1.1.1.2

Subtract $4 y$ from both sides of the equation.

$- x + 1 = - y^{2} - 4 y$

Step 1.1.1.3

Subtract $1$ from both sides of the equation.

$- x = - y^{2} - 4 y - 1$

Step 1.1.2

Divide each term in $- x = - y^{2} - 4 y - 1$ by $- 1$ and simplify.

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Step 1.1.2.1

Divide each term in $- x = - y^{2} - 4 y - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- y^{2}}{- 1} + \frac{- 4 y}{- 1} + \frac{- 1}{- 1}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- y^{2}}{- 1} + \frac{- 4 y}{- 1} + \frac{- 1}{- 1}$

Step 1.1.2.2.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 1} + \frac{- 4 y}{- 1} + \frac{- 1}{- 1}$

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

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{1} + \frac{- 4 y}{- 1} + \frac{- 1}{- 1}$

Step 1.1.2.3.1.2

Divide $y^{2}$ by $1$ .

$x = y^{2} + \frac{- 4 y}{- 1} + \frac{- 1}{- 1}$

Step 1.1.2.3.1.3

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

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

Step 1.1.2.3.1.4

Rewrite $- 1 \cdot (- 4 y)$ as $- (- 4 y)$ .

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

Step 1.1.2.3.1.5

Multiply $- 4$ by $- 1$ .

$x = y^{2} + 4 y + \frac{- 1}{- 1}$

Step 1.1.2.3.1.6

Divide $- 1$ by $- 1$ .

$x = y^{2} + 4 y + 1$

Step 1.2

Complete the square for $y^{2} + 4 y + 1$ .

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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 $c$ .

$a = 1$

$b = 4$

$c = 1$

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

$d = \frac{4}{2 \cdot 1}$

Step 1.2.3.2

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

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Step 1.2.3.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.2

Cancel the common factors.

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Step 1.2.3.2.2.1

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

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

Step 1.2.3.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

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

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 1.2.4.2.1.1.2

Cancel the common factors.

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Step 1.2.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 1$ .

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

Step 1.2.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 1 - 1 \cdot 4$

Step 1.2.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 1 - 4$

Step 1.2.4.2.2

Subtract $4$ from $1$ .

$e = - 3$

Step 1.2.5

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = 1$

$h = - 3$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(- 3, - 2)$

Step 4