Pre-Algebra Examples

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

Step 1

Rewrite the function as an equation.

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 2.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.1.3.2

Subtract $x$ from $- x$ .

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

Step 2.1.4

Apply the distributive property.

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

Step 2.1.5

Simplify.

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Step 2.1.5.1

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

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

Step 2.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.5.2.1

Factor $2$ out of $4$ .

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

Step 2.1.5.2.2

Factor $2$ out of $- 2 x$ .

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

Step 2.1.5.2.3

Cancel the common factor.

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

Step 2.1.5.2.4

Rewrite the expression.

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Multiply $\frac{1}{4}$ by $1$ .

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

Step 2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 2$ by $4$ .

$y = \frac{x^{2}}{4} - \frac{x}{2} + \frac{1 - 8}{4}$

Step 2.5.2

Subtract $8$ from $1$ .

$y = \frac{x^{2}}{4} - \frac{x}{2} + \frac{- 7}{4}$

Step 2.6

Move the negative in front of the fraction.

$y = \frac{x^{2}}{4} - \frac{x}{2} - \frac{7}{4}$