Algebra Examples

$f (x) = - \frac{1}{3} {(x - 2)}^{2} + 2$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$f (x) = a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

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

$f (x) = - \frac{1}{3} \cdot ((x - 2) (x - 2)) + 2$

Step 2.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$f (x) = - \frac{1}{3} \cdot (x (x - 2) - 2 (x - 2)) + 2$

Step 2.2.2

Apply the distributive property.

$f (x) = - \frac{1}{3} \cdot (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 2$

Step 2.2.3

Apply the distributive property.

$f (x) = - \frac{1}{3} \cdot (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 2$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 2.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 2.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify.

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Step 2.5.1

Combine $x^{2}$ and $\frac{1}{3}$ .

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

Step 2.5.2

Multiply $- \frac{1}{3} (- 4 x)$ .

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Step 2.5.2.1

Multiply $- 4$ by $- 1$ .

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

Step 2.5.2.2

Combine $4$ and $\frac{1}{3}$ .

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

Step 2.5.2.3

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

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

Step 2.5.3

Multiply $- \frac{1}{3} \cdot 4$ .

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Step 2.5.3.1

Multiply $4$ by $- 1$ .

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

Step 2.5.3.2

Combine $- 4$ and $\frac{1}{3}$ .

$f (x) = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{- 4}{3} + 2$

Step 2.6

Move the negative in front of the fraction.

$f (x) = - \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{4}{3} + 2$

Step 3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $2$ by $3$ .

$f (x) = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{- 4 + 6}{3}$

Step 6.2

Add $- 4$ and $6$ .

$f (x) = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{2}{3}$

Step 7

Reorder terms.

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

Step 8

Remove parentheses.

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