Pre-Algebra Examples

$f (x) = - \frac{3}{4} \cdot {(x - 1)}^{2} - 5$

Step 1

Rewrite the function as an equation.

$y = - \frac{3}{4} \cdot {(x - 1)}^{2} - 5$

Step 2

Simplify $- \frac{3}{4} \cdot {(x - 1)}^{2} - 5$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$y = - \frac{3}{4} \cdot ((x - 1) (x - 1)) - 5$

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.

$y = - \frac{3}{4} \cdot (x (x - 1) - 1 (x - 1)) - 5$

Step 2.1.2.2

Apply the distributive property.

$y = - \frac{3}{4} \cdot (x \cdot x + x \cdot - 1 - 1 (x - 1)) - 5$

Step 2.1.2.3

Apply the distributive property.

$y = - \frac{3}{4} \cdot (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - 5$

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 $x$ .

$y = - \frac{3}{4} \cdot (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - 5$

Step 2.1.3.1.2

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

$y = - \frac{3}{4} \cdot (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - 5$

Step 2.1.3.1.3

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

$y = - \frac{3}{4} \cdot (x^{2} - x - 1 x - 1 \cdot - 1) - 5$

Step 2.1.3.1.4

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

$y = - \frac{3}{4} \cdot (x^{2} - x - x - 1 \cdot - 1) - 5$

Step 2.1.3.1.5

Multiply $- 1$ by $- 1$ .

$y = - \frac{3}{4} \cdot (x^{2} - x - x + 1) - 5$

Step 2.1.3.2

Subtract $x$ from $- x$ .

$y = - \frac{3}{4} \cdot (x^{2} - 2 x + 1) - 5$

Step 2.1.4

Apply the distributive property.

$y = - \frac{3}{4} x^{2} - \frac{3}{4} (- 2 x) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Combine $x^{2}$ and $\frac{3}{4}$ .

$y = - \frac{x^{2} \cdot 3}{4} - \frac{3}{4} (- 2 x) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.5.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y = - \frac{x^{2} \cdot 3}{4} + \frac{- 3}{4} (- 2 x) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.2.2

Factor $2$ out of $4$ .

$y = - \frac{x^{2} \cdot 3}{4} + \frac{- 3}{2 (2)} (- 2 x) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.2.3

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

$y = - \frac{x^{2} \cdot 3}{4} + \frac{- 3}{2 (2)} (2 (- x)) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.2.4

Cancel the common factor.

$y = - \frac{x^{2} \cdot 3}{4} + \frac{- 3}{2 \cdot 2} (2 (- x)) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.2.5

Rewrite the expression.

$y = - \frac{x^{2} \cdot 3}{4} + \frac{- 3}{2} (- x) - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.3

Combine $\frac{- 3}{2}$ and $x$ .

$y = - \frac{x^{2} \cdot 3}{4} - \frac{- 3 x}{2} - \frac{3}{4} \cdot 1 - 5$

Step 2.1.5.4

Multiply $- 1$ by $1$ .

$y = - \frac{x^{2} \cdot 3}{4} - \frac{- 3 x}{2} - \frac{3}{4} - 5$

Step 2.1.6

Simplify each term.

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Step 2.1.6.1

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 \cdot x^{2}}{4} - \frac{- 3 x}{2} - \frac{3}{4} - 5$

Step 2.1.6.2

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{4} - - \frac{3 x}{2} - \frac{3}{4} - 5$

Step 2.1.6.3

Multiply $- - \frac{3 x}{2}$ .

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Step 2.1.6.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{3 x^{2}}{4} + 1 \frac{3 x}{2} - \frac{3}{4} - 5$

Step 2.1.6.3.2

Multiply $\frac{3 x}{2}$ by $1$ .

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} - \frac{3}{4} - 5$

Step 2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} - \frac{3}{4} - 5 \cdot \frac{4}{4}$

Step 2.3

Combine $- 5$ and $\frac{4}{4}$ .

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} - \frac{3}{4} + \frac{- 5 \cdot 4}{4}$

Step 2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} + \frac{- 3 - 5 \cdot 4}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 5$ by $4$ .

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} + \frac{- 3 - 20}{4}$

Step 2.5.2

Subtract $20$ from $- 3$ .

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} + \frac{- 23}{4}$

Step 2.6

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{4} + \frac{3 x}{2} - \frac{23}{4}$