Precalculus Examples

$x^{2} + 6 x - 2 y + 5 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

$6 x - 2 y + 5 = - x^{2}$

Step 1.1.1.2

Subtract $6 x$ from both sides of the equation.

$- 2 y + 5 = - x^{2} - 6 x$

Step 1.1.1.3

Subtract $5$ from both sides of the equation.

$- 2 y = - x^{2} - 6 x - 5$

Step 1.1.2

Divide each term in $- 2 y = - x^{2} - 6 x - 5$ by $- 2$ and simplify.

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Step 1.1.2.1

Divide each term in $- 2 y = - x^{2} - 6 x - 5$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- x^{2}}{- 2} + \frac{- 6 x}{- 2} + \frac{- 5}{- 2}$

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.

$\frac{- 2 y}{- 2} = \frac{- x^{2}}{- 2} + \frac{- 6 x}{- 2} + \frac{- 5}{- 2}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 2} + \frac{- 6 x}{- 2} + \frac{- 5}{- 2}$

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

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{2} + \frac{- 6 x}{- 2} + \frac{- 5}{- 2}$

Step 1.1.2.3.1.2

Cancel the common factor of $- 6$ and $- 2$ .

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Step 1.1.2.3.1.2.1

Factor $- 2$ out of $- 6 x$ .

$y = \frac{x^{2}}{2} + \frac{- 2 (3 x)}{- 2} + \frac{- 5}{- 2}$

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 $- 2$ .

$y = \frac{x^{2}}{2} + \frac{- 2 (3 x)}{- 2 (1)} + \frac{- 5}{- 2}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{2} + \frac{- 2 (3 x)}{- 2 \cdot 1} + \frac{- 5}{- 2}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{2} + \frac{3 x}{1} + \frac{- 5}{- 2}$

Step 1.1.2.3.1.2.2.4

Divide $3 x$ by $1$ .

$y = \frac{x^{2}}{2} + 3 x + \frac{- 5}{- 2}$

Step 1.1.2.3.1.3

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{2} + 3 x + \frac{5}{2}$

Step 1.2

Complete the square for $\frac{x^{2}}{2} + 3 x + \frac{5}{2}$ .

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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 $c$ .

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

$b = 3$

$c = \frac{5}{2}$

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

$d = \frac{3}{2 (\frac{1}{2})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

$d = \frac{3}{\frac{2}{2}}$

Step 1.2.3.2.2

Divide $2$ by $2$ .

$d = \frac{3}{1}$

Step 1.2.3.2.3

Divide $3$ by $1$ .

$d = 3$

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

$e = \frac{5}{2} - \frac{3^{2}}{4 (\frac{1}{2})}$

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

Raise $3$ to the power of $2$ .

$e = \frac{5}{2} - \frac{9}{4 (\frac{1}{2})}$

Step 1.2.4.2.1.2

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

$e = \frac{5}{2} - \frac{9}{\frac{4}{2}}$

Step 1.2.4.2.1.3

Divide $4$ by $2$ .

$e = \frac{5}{2} - \frac{9}{2}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{5 - 9}{2}$

Step 1.2.4.2.3

Subtract $9$ from $5$ .

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

Step 1.2.4.2.4

Divide $- 4$ by $2$ .

$e = - 2$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , 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