Pre-Algebra Examples

$f (x) = 9 {(x - \frac{1}{6})}^{2} - 4$

Step 1

Rewrite the function as an equation.

$y = 9 {(x - \frac{1}{6})}^{2} - 4$

Step 2

Simplify $9 {(x - \frac{1}{6})}^{2} - 4$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - \frac{1}{6})}^{2}$ as $(x - \frac{1}{6}) (x - \frac{1}{6})$ .

$y = 9 ((x - \frac{1}{6}) (x - \frac{1}{6})) - 4$

Step 2.1.2

Expand $(x - \frac{1}{6}) (x - \frac{1}{6})$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$y = 9 (x (x - \frac{1}{6}) - \frac{1}{6} (x - \frac{1}{6})) - 4$

Step 2.1.2.2

Apply the distributive property.

$y = 9 (x \cdot x + x (- \frac{1}{6}) - \frac{1}{6} (x - \frac{1}{6})) - 4$

Step 2.1.2.3

Apply the distributive property.

$y = 9 (x \cdot x + x (- \frac{1}{6}) - \frac{1}{6} x - \frac{1}{6} (- \frac{1}{6})) - 4$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$y = 9 (x^{2} + x (- \frac{1}{6}) - \frac{1}{6} x - \frac{1}{6} (- \frac{1}{6})) - 4$

Step 2.1.3.1.2

Combine $x$ and $\frac{1}{6}$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{1}{6} x - \frac{1}{6} (- \frac{1}{6})) - 4$

Step 2.1.3.1.3

Combine $x$ and $\frac{1}{6}$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{x}{6} - \frac{1}{6} (- \frac{1}{6})) - 4$

Step 2.1.3.1.4

Multiply $- \frac{1}{6} (- \frac{1}{6})$ .

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Step 2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{x}{6} + 1 (\frac{1}{6}) \frac{1}{6}) - 4$

Step 2.1.3.1.4.2

Multiply $\frac{1}{6}$ by $1$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{x}{6} + \frac{1}{6} \cdot \frac{1}{6}) - 4$

Step 2.1.3.1.4.3

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{x}{6} + \frac{1}{6 \cdot 6}) - 4$

Step 2.1.3.1.4.4

Multiply $6$ by $6$ .

$y = 9 (x^{2} - \frac{x}{6} - \frac{x}{6} + \frac{1}{36}) - 4$

Step 2.1.3.2

Subtract $\frac{x}{6}$ from $- \frac{x}{6}$ .

$y = 9 (x^{2} - 2 \frac{x}{6} + \frac{1}{36}) - 4$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

Cancel the common factor of $2$ .

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Step 2.1.4.1.1

Factor $2$ out of $- 2$ .

$y = 9 (x^{2} + 2 (- 1) \frac{x}{6} + \frac{1}{36}) - 4$

Step 2.1.4.1.2

Factor $2$ out of $6$ .

$y = 9 (x^{2} + 2 \cdot - 1 \frac{x}{2 \cdot 3} + \frac{1}{36}) - 4$

Step 2.1.4.1.3

Cancel the common factor.

$y = 9 (x^{2} + 2 \cdot - 1 \frac{x}{2 \cdot 3} + \frac{1}{36}) - 4$

Step 2.1.4.1.4

Rewrite the expression.

$y = 9 (x^{2} - 1 \frac{x}{3} + \frac{1}{36}) - 4$

Step 2.1.4.2

Rewrite $- 1 \frac{x}{3}$ as $- \frac{x}{3}$ .

$y = 9 (x^{2} - \frac{x}{3} + \frac{1}{36}) - 4$

Step 2.1.5

Apply the distributive property.

$y = 9 x^{2} + 9 (- \frac{x}{3}) + 9 (\frac{1}{36}) - 4$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Cancel the common factor of $3$ .

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Step 2.1.6.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$y = 9 x^{2} + 9 \frac{- x}{3} + 9 (\frac{1}{36}) - 4$

Step 2.1.6.1.2

Factor $3$ out of $9$ .

$y = 9 x^{2} + 3 (3) \frac{- x}{3} + 9 (\frac{1}{36}) - 4$

Step 2.1.6.1.3

Cancel the common factor.

$y = 9 x^{2} + 3 \cdot 3 \frac{- x}{3} + 9 (\frac{1}{36}) - 4$

Step 2.1.6.1.4

Rewrite the expression.

$y = 9 x^{2} + 3 (- x) + 9 (\frac{1}{36}) - 4$

Step 2.1.6.2

Multiply $- 1$ by $3$ .

$y = 9 x^{2} - 3 x + 9 (\frac{1}{36}) - 4$

Step 2.1.6.3

Cancel the common factor of $9$ .

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Step 2.1.6.3.1

Factor $9$ out of $36$ .

$y = 9 x^{2} - 3 x + 9 \frac{1}{9 (4)} - 4$

Step 2.1.6.3.2

Cancel the common factor.

$y = 9 x^{2} - 3 x + 9 \frac{1}{9 \cdot 4} - 4$

Step 2.1.6.3.3

Rewrite the expression.

$y = 9 x^{2} - 3 x + \frac{1}{4} - 4$

Step 2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 9 x^{2} - 3 x + \frac{1}{4} - 4 \cdot \frac{4}{4}$

Step 2.3

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

$y = 9 x^{2} - 3 x + \frac{1}{4} + \frac{- 4 \cdot 4}{4}$

Step 2.4

Combine the numerators over the common denominator.

$y = 9 x^{2} - 3 x + \frac{1 - 4 \cdot 4}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 4$ by $4$ .

$y = 9 x^{2} - 3 x + \frac{1 - 16}{4}$

Step 2.5.2

Subtract $16$ from $1$ .

$y = 9 x^{2} - 3 x + \frac{- 15}{4}$

Step 2.6

Move the negative in front of the fraction.

$y = 9 x^{2} - 3 x - \frac{15}{4}$