Algebra Examples

$h (x) = - \frac{1}{64} x^{2} + \frac{13}{32} x + 2$

Step 1

Write $h (x) = - \frac{1}{64} x^{2} + \frac{13}{32} x + 2$ as an equation.

$y = - \frac{1}{64} x^{2} + \frac{13}{32} x + 2$

Step 2

Simplify each term.

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Step 2.1

Combine $x^{2}$ and $\frac{1}{64}$ .

$y = - \frac{x^{2}}{64} + \frac{13}{32} x + 2$

Step 2.2

Combine $\frac{13}{32}$ and $x$ .

$y = - \frac{x^{2}}{64} + \frac{13 x}{32} + 2$

Step 3

Complete the square for $- \frac{x^{2}}{64} + \frac{13 x}{32} + 2$ .

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Step 3.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{64}$

$b = \frac{13}{32}$

$c = 2$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{13}{32}}{2 (- \frac{1}{64})}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{13}{32} \cdot \frac{1}{2 (- \frac{1}{64})}$

Step 3.3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 3.3.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{13}{32} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{64})}$

Step 3.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{13}{32} (- \frac{1}{2 (\frac{1}{64})})$

Step 3.3.2.3

Combine $2$ and $\frac{1}{64}$ .

$d = \frac{13}{32} (- \frac{1}{\frac{2}{64}})$

Step 3.3.2.4

Cancel the common factor of $2$ and $64$ .

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Step 3.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{13}{32} (- \frac{1}{\frac{2 (1)}{64}})$

Step 3.3.2.4.2

Cancel the common factors.

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Step 3.3.2.4.2.1

Factor $2$ out of $64$ .

$d = \frac{13}{32} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 32}})$

Step 3.3.2.4.2.2

Cancel the common factor.

$d = \frac{13}{32} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 32}})$

Step 3.3.2.4.2.3

Rewrite the expression.

$d = \frac{13}{32} (- \frac{1}{\frac{1}{32}})$

Step 3.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{13}{32} (- (1 \cdot 32))$

Step 3.3.2.6

Cancel the common factor of $32$ .

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Step 3.3.2.6.1

Factor $32$ out of $- (1 \cdot 32)$ .

$d = \frac{13}{32} (32 (- 1 \cdot (1)))$

Step 3.3.2.6.2

Cancel the common factor.

$d = \frac{13}{32} (32 (- 1 \cdot (1)))$

Step 3.3.2.6.3

Rewrite the expression.

$d = 13 (- 1 \cdot (1))$

Step 3.3.2.7

Multiply $- 1$ by $1$ .

$d = 13 \cdot - 1$

Step 3.3.2.8

Multiply $13$ by $- 1$ .

$d = - 13$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 2 - \frac{{(\frac{13}{32})}^{2}}{4 (- \frac{1}{64})}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Apply the product rule to $\frac{13}{32}$ .

$e = 2 - \frac{\frac{13^{2}}{32^{2}}}{4 (- \frac{1}{64})}$

Step 3.4.2.1.1.2

Raise $13$ to the power of $2$ .

$e = 2 - \frac{\frac{169}{32^{2}}}{4 (- \frac{1}{64})}$

Step 3.4.2.1.1.3

Raise $32$ to the power of $2$ .

$e = 2 - \frac{\frac{169}{1024}}{4 (- \frac{1}{64})}$

Step 3.4.2.1.2

Simplify the denominator.

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Step 3.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 2 - \frac{\frac{169}{1024}}{- 4 (\frac{1}{64})}$

Step 3.4.2.1.2.2

Combine $- 4$ and $\frac{1}{64}$ .

$e = 2 - \frac{\frac{169}{1024}}{\frac{- 4}{64}}$

Step 3.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.4.2.1.3.1

Cancel the common factor of $- 4$ and $64$ .

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Step 3.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = 2 - \frac{\frac{169}{1024}}{\frac{4 (- 1)}{64}}$

Step 3.4.2.1.3.1.2

Cancel the common factors.

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Step 3.4.2.1.3.1.2.1

Factor $4$ out of $64$ .

$e = 2 - \frac{\frac{169}{1024}}{\frac{4 \cdot - 1}{4 \cdot 16}}$

Step 3.4.2.1.3.1.2.2

Cancel the common factor.

$e = 2 - \frac{\frac{169}{1024}}{\frac{4 \cdot - 1}{4 \cdot 16}}$

Step 3.4.2.1.3.1.2.3

Rewrite the expression.

$e = 2 - \frac{\frac{169}{1024}}{\frac{- 1}{16}}$

Step 3.4.2.1.3.2

Move the negative in front of the fraction.

$e = 2 - \frac{\frac{169}{1024}}{- \frac{1}{16}}$

Step 3.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 2 - (\frac{169}{1024} (- 1 \cdot 16))$

Step 3.4.2.1.5

Cancel the common factor of $16$ .

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Step 3.4.2.1.5.1

Factor $16$ out of $1024$ .

$e = 2 - (\frac{169}{16 (64)} (- 1 \cdot 16))$

Step 3.4.2.1.5.2

Factor $16$ out of $- 1 \cdot 16$ .

$e = 2 - (\frac{169}{16 (64)} (16 \cdot - 1))$

Step 3.4.2.1.5.3

Cancel the common factor.

$e = 2 - (\frac{169}{16 \cdot 64} (16 \cdot - 1))$

Step 3.4.2.1.5.4

Rewrite the expression.

$e = 2 - (\frac{169}{64} \cdot - 1)$

Step 3.4.2.1.6

Combine $\frac{169}{64}$ and $- 1$ .

$e = 2 - \frac{169 \cdot - 1}{64}$

Step 3.4.2.1.7

Multiply $169$ by $- 1$ .

$e = 2 - \frac{- 169}{64}$

Step 3.4.2.1.8

Move the negative in front of the fraction.

$e = 2 - - \frac{169}{64}$

Step 3.4.2.1.9

Multiply $- - \frac{169}{64}$ .

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Step 3.4.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{169}{64})$

Step 3.4.2.1.9.2

Multiply $\frac{169}{64}$ by $1$ .

$e = 2 + \frac{169}{64}$

Step 3.4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$e = 2 \cdot \frac{64}{64} + \frac{169}{64}$

Step 3.4.2.3

Combine $2$ and $\frac{64}{64}$ .

$e = \frac{2 \cdot 64}{64} + \frac{169}{64}$

Step 3.4.2.4

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 64 + 169}{64}$

Step 3.4.2.5

Simplify the numerator.

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Step 3.4.2.5.1

Multiply $2$ by $64$ .

$e = \frac{128 + 169}{64}$

Step 3.4.2.5.2

Add $128$ and $169$ .

$e = \frac{297}{64}$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{64} {(x - 13)}^{2} + \frac{297}{64}$ .

$- \frac{1}{64} {(x - 13)}^{2} + \frac{297}{64}$

Step 4

Set $y$ equal to the new right side.

$y = - \frac{1}{64} \cdot {(x - 13)}^{2} + \frac{297}{64}$