Precalculus Examples

$x^{2} + 6 x = 4 y - 5$

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 $4 y - 5 = x^{2} + 6 x$ .

$4 y - 5 = x^{2} + 6 x$

Step 1.1.2

Add $5$ to both sides of the equation.

$4 y = x^{2} + 6 x + 5$

Step 1.1.3

Divide each term in $4 y = x^{2} + 6 x + 5$ by $4$ and simplify.

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Step 1.1.3.1

Divide each term in $4 y = x^{2} + 6 x + 5$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{6 x}{4} + \frac{5}{4}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $4$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{6 x}{4} + \frac{5}{4}$

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{6 x}{4} + \frac{5}{4}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $6$ and $4$ .

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Step 1.1.3.3.1.1

Factor $2$ out of $6 x$ .

$y = \frac{x^{2}}{4} + \frac{2 (3 x)}{4} + \frac{5}{4}$

Step 1.1.3.3.1.2

Cancel the common factors.

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Step 1.1.3.3.1.2.1

Factor $2$ out of $4$ .

$y = \frac{x^{2}}{4} + \frac{2 (3 x)}{2 (2)} + \frac{5}{4}$

Step 1.1.3.3.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{4} + \frac{2 (3 x)}{2 \cdot 2} + \frac{5}{4}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{3 x}{2} + \frac{5}{4}$

Step 1.2

Complete the square for $\frac{x^{2}}{4} + \frac{3 x}{2} + \frac{5}{4}$ .

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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 $c$ .

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

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

$c = \frac{5}{4}$

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

$d = \frac{\frac{3}{2}}{2 (\frac{1}{4})}$

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.

$d = \frac{3}{2} \cdot \frac{1}{2 (\frac{1}{4})}$

Step 1.2.3.2.2

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

$d = \frac{3}{2} \cdot \frac{1}{\frac{2}{4}}$

Step 1.2.3.2.3

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

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 (1)}{4}}$

Step 1.2.3.2.3.2

Cancel the common factors.

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Step 1.2.3.2.3.2.1

Factor $2$ out of $4$ .

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = \frac{3}{2} \cdot \frac{1}{\frac{1}{2}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.5

Cancel the common factor of $2$ .

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Step 1.2.3.2.5.1

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

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

Step 1.2.3.2.5.2

Cancel the common factor.

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

Step 1.2.3.2.5.3

Rewrite the expression.

$d = 3$

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

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

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 $\frac{3}{2}$ .

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

Step 1.2.4.2.1.1.2

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

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

Step 1.2.4.2.1.1.3

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

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

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

Step 1.2.4.2.1.4

Divide $\frac{9}{4}$ by $1$ .

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

Step 1.2.4.2.2

Combine the numerators over the common denominator.

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

Step 1.2.4.2.3

Subtract $9$ from $5$ .

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

Step 1.2.4.2.4

Divide $- 4$ by $4$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x + 3)}^{2} - 1$ .

$\frac{1}{4} {(x + 3)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{4} \cdot {(x + 3)}^{2} - 1$

Step 2

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

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

$h = - 3$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(- 3, - 1)$

Step 4