Precalculus Examples

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

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

Rewrite the equation as $2 y = x^{2} - 4 x$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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.

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

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

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

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Step 1.1.2.3.1.1

Factor $2$ out of $- 4 x$ .

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

Step 1.1.2.3.1.2

Cancel the common factors.

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Step 1.1.2.3.1.2.1

Factor $2$ out of $2$ .

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

Step 1.1.2.3.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.2.4

Divide $- 2 x$ by $1$ .

$y = \frac{x^{2}}{2} - 2 x$

Step 1.2

Complete the square for $\frac{x^{2}}{2} - 2 x$ .

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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 $c$ .

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

$b = - 2$

$c = 0$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Step 1.2.3.2.3

Multiply $- 1$ by $2$ .

$d = - 2$

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

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

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

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

Step 1.2.4.2.1.2

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

$e = 0 - \frac{4}{\frac{4}{2}}$

Step 1.2.4.2.1.3

Divide $4$ by $2$ .

$e = 0 - \frac{4}{2}$

Step 1.2.4.2.1.4

Divide $4$ by $2$ .

$e = 0 - 1 \cdot 2$

Step 1.2.4.2.1.5

Multiply $- 1$ by $2$ .

$e = 0 - 2$

Step 1.2.4.2.2

Subtract $2$ from $0$ .

$e = - 2$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 2$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(2, - 2)$

Step 4