Algebra Examples

$f (x) = \frac{1}{16} x^{2} - 8 x + 11$

Step 1

Write $f (x) = \frac{1}{16} x^{2} - 8 x + 11$ as an equation.

$y = \frac{1}{16} x^{2} - 8 x + 11$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{1}{16} x^{2} - 8 x + 11 = y$

Step 3

Subtract $11$ from both sides of the equation.

$\frac{1}{16} x^{2} - 8 x = y - 11$

Step 4

Complete the square.

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Step 4.1

Combine $\frac{1}{16}$ and $x^{2}$ .

$\frac{x^{2}}{16} - 8 x = y - 11$

Step 4.2

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

$a = \frac{1}{16}$

$b = - 8$

$c = 0 = y - 11$

Step 4.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 11$

Step 4.4

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

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Step 4.4.1

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

$d = \frac{- 8}{2 (\frac{1}{16})}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 (\frac{1}{16})}$

Step 4.4.2.1.2

Cancel the common factors.

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Step 4.4.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 (\frac{1}{16})}$

Step 4.4.2.1.2.2

Rewrite the expression.

$d = \frac{- 4}{\frac{1}{16}}$

Step 4.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 4 \cdot 16$

Step 4.4.2.3

Multiply $- 4$ by $16$ .

$d = - 64$

Step 4.5

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

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Step 4.5.1

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

$e = 0 - \frac{{(- 8)}^{2}}{4 (\frac{1}{16})}$

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 (\frac{1}{16})}$

Step 4.5.2.1.2

Combine $4$ and $\frac{1}{16}$ .

$e = 0 - \frac{64}{\frac{4}{16}}$

Step 4.5.2.1.3

Cancel the common factor of $4$ and $16$ .

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Step 4.5.2.1.3.1

Factor $4$ out of $4$ .

$e = 0 - \frac{64}{\frac{4 (1)}{16}}$

Step 4.5.2.1.3.2

Cancel the common factors.

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Step 4.5.2.1.3.2.1

Factor $4$ out of $16$ .

$e = 0 - \frac{64}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 4.5.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{64}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 4.5.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{64}{\frac{1}{4}}$

Step 4.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (64 \cdot 4)$

Step 4.5.2.1.5

Multiply $- (64 \cdot 4)$ .

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Step 4.5.2.1.5.1

Multiply $64$ by $4$ .

$e = 0 - 1 \cdot 256$

Step 4.5.2.1.5.2

Multiply $- 1$ by $256$ .

$e = 0 - 256$

Step 4.5.2.2

Subtract $256$ from $0$ .

$e = - 256$

Step 4.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{16} {(x - 64)}^{2} - 256$ .

$\frac{1}{16} \cdot {(x - 64)}^{2} - 256 = y - 11$

Step 5

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

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Step 5.1

Add $256$ to both sides of the equation.

$\frac{1}{16} \cdot {(x - 64)}^{2} = y - 11 + 256$

Step 5.2

Add $- 11$ and $256$ .

$\frac{1}{16} \cdot {(x - 64)}^{2} = y + 245$

Step 6

Multiply each term in $\frac{1}{16} \cdot {(x - 64)}^{2} = y + 245$ by $16$ to eliminate the fractions.

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Step 6.1

Multiply each term in $\frac{1}{16} \cdot {(x - 64)}^{2} = y + 245$ by $16$ .

$\frac{1}{16} \cdot {(x - 64)}^{2} \cdot 16 = y \cdot 16 + 245 \cdot 16$

Step 6.2

Simplify the left side.

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Step 6.2.1

Combine $\frac{1}{16}$ and ${(x - 64)}^{2}$ .

$\frac{{(x - 64)}^{2}}{16} \cdot 16 = y \cdot 16 + 245 \cdot 16$

Step 6.2.2

Cancel the common factor of $16$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{{(x - 64)}^{2}}{16} \cdot 16 = y \cdot 16 + 245 \cdot 16$

Step 6.2.2.2

Rewrite the expression.

${(x - 64)}^{2} = y \cdot 16 + 245 \cdot 16$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Move $16$ to the left of $y$ .

${(x - 64)}^{2} = 16 \cdot y + 245 \cdot 16$

Step 6.3.1.2

Multiply $245$ by $16$ .

${(x - 64)}^{2} = 16 y + 3920$

Step 7

Factor $16$ out of $16 y + 3920$ .

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Step 7.1

Factor $16$ out of $16 y$ .

${(x - 64)}^{2} = 16 (y) + 3920$

Step 7.2

Factor $16$ out of $3920$ .

${(x - 64)}^{2} = 16 y + 16 \cdot 245$

Step 7.3

Factor $16$ out of $16 y + 16 \cdot 245$ .

${(x - 64)}^{2} = 16 (y + 245)$