Precalculus Examples

$x^{2} + 10 x = 8 y - 17$

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 $8 y - 17 = x^{2} + 10 x$ .

$8 y - 17 = x^{2} + 10 x$

Step 1.1.2

Add $17$ to both sides of the equation.

$8 y = x^{2} + 10 x + 17$

Step 1.1.3

Divide each term in $8 y = x^{2} + 10 x + 17$ by $8$ and simplify.

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Step 1.1.3.1

Divide each term in $8 y = x^{2} + 10 x + 17$ by $8$ .

$\frac{8 y}{8} = \frac{x^{2}}{8} + \frac{10 x}{8} + \frac{17}{8}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $8$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x^{2}}{8} + \frac{10 x}{8} + \frac{17}{8}$

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{8} + \frac{10 x}{8} + \frac{17}{8}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $10$ and $8$ .

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Step 1.1.3.3.1.1

Factor $2$ out of $10 x$ .

$y = \frac{x^{2}}{8} + \frac{2 (5 x)}{8} + \frac{17}{8}$

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 $8$ .

$y = \frac{x^{2}}{8} + \frac{2 (5 x)}{2 (4)} + \frac{17}{8}$

Step 1.1.3.3.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{2 (5 x)}{2 \cdot 4} + \frac{17}{8}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{5 x}{4} + \frac{17}{8}$

Step 1.2

Complete the square for $\frac{x^{2}}{8} + \frac{5 x}{4} + \frac{17}{8}$ .

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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 $c$ .

$a = \frac{1}{8}$

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

$c = \frac{17}{8}$

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

$d = \frac{\frac{5}{4}}{2 (\frac{1}{8})}$

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.

$d = \frac{5}{4} \cdot \frac{1}{2 (\frac{1}{8})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{5}{4} \cdot \frac{1}{\frac{2}{8}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $8$ .

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = \frac{5}{4} \cdot \frac{1}{\frac{2 (1)}{8}}$

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 $8$ .

$d = \frac{5}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = \frac{5}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = \frac{5}{4} \cdot \frac{1}{\frac{1}{4}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{4} (1 \cdot 4)$

Step 1.2.3.2.5

Cancel the common factor of $4$ .

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Step 1.2.3.2.5.1

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

$d = \frac{5}{4} (4 \cdot 1)$

Step 1.2.3.2.5.2

Cancel the common factor.

$d = \frac{5}{4} (4 \cdot 1)$

Step 1.2.3.2.5.3

Rewrite the expression.

$d = 5$

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

$e = \frac{17}{8} - \frac{{(\frac{5}{4})}^{2}}{4 (\frac{1}{8})}$

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 $\frac{5}{4}$ .

$e = \frac{17}{8} - \frac{\frac{5^{2}}{4^{2}}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.1.2

Raise $5$ to the power of $2$ .

$e = \frac{17}{8} - \frac{\frac{25}{4^{2}}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.1.3

Raise $4$ to the power of $2$ .

$e = \frac{17}{8} - \frac{\frac{25}{16}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{8}$ .

$e = \frac{17}{8} - \frac{\frac{25}{16}}{\frac{4}{8}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $8$ .

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{17}{8} - \frac{\frac{25}{16}}{\frac{4 (1)}{8}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $8$ .

$e = \frac{17}{8} - \frac{\frac{25}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{17}{8} - \frac{\frac{25}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{17}{8} - \frac{\frac{25}{16}}{\frac{1}{2}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{17}{8} - (\frac{25}{16} \cdot 2)$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Factor $2$ out of $16$ .

$e = \frac{17}{8} - (\frac{25}{2 (8)} \cdot 2)$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = \frac{17}{8} - (\frac{25}{2 \cdot 8} \cdot 2)$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = \frac{17}{8} - \frac{25}{8}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{17 - 25}{8}$

Step 1.2.4.2.3

Subtract $25$ from $17$ .

$e = \frac{- 8}{8}$

Step 1.2.4.2.4

Divide $- 8$ by $8$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x + 5)}^{2} - 1$ .

$\frac{1}{8} {(x + 5)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{8} \cdot {(x + 5)}^{2} - 1$

Step 2

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

$a = \frac{1}{8}$

$h = - 5$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(- 5, - 1)$

Step 4