Algebra Examples

$f (x) = - \frac{2}{9} {(x + 3)}^{2} + 2$

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 ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$f (x) = - \frac{2}{9} \cdot ((x + 3) (x + 3)) + 2$

Step 2.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$f (x) = - \frac{2}{9} \cdot (x (x + 3) + 3 (x + 3)) + 2$

Step 2.2.2

Apply the distributive property.

$f (x) = - \frac{2}{9} \cdot (x \cdot x + x \cdot 3 + 3 (x + 3)) + 2$

Step 2.2.3

Apply the distributive property.

$f (x) = - \frac{2}{9} \cdot (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 2$

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 $x$ .

$f (x) = - \frac{2}{9} \cdot (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 2$

Step 2.3.1.2

Move $3$ to the left of $x$ .

$f (x) = - \frac{2}{9} \cdot (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 2$

Step 2.3.1.3

Multiply $3$ by $3$ .

$f (x) = - \frac{2}{9} \cdot (x^{2} + 3 x + 3 x + 9) + 2$

Step 2.3.2

Add $3 x$ and $3 x$ .

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify.

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Step 2.5.1

Combine $x^{2}$ and $\frac{2}{9}$ .

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

Step 2.5.2

Cancel the common factor of $3$ .

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Step 2.5.2.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

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

Step 2.5.2.2

Factor $3$ out of $9$ .

$f (x) = - \frac{x^{2} \cdot 2}{9} + \frac{- 2}{3 (3)} \cdot (6 x) - \frac{2}{9} \cdot 9 + 2$

Step 2.5.2.3

Factor $3$ out of $6 x$ .

$f (x) = - \frac{x^{2} \cdot 2}{9} + \frac{- 2}{3 (3)} \cdot (3 (2 x)) - \frac{2}{9} \cdot 9 + 2$

Step 2.5.2.4

Cancel the common factor.

$f (x) = - \frac{x^{2} \cdot 2}{9} + \frac{- 2}{3 \cdot 3} \cdot (3 (2 x)) - \frac{2}{9} \cdot 9 + 2$

Step 2.5.2.5

Rewrite the expression.

$f (x) = - \frac{x^{2} \cdot 2}{9} + \frac{- 2}{3} \cdot (2 x) - \frac{2}{9} \cdot 9 + 2$

Step 2.5.3

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

$f (x) = - \frac{x^{2} \cdot 2}{9} + \frac{2 \cdot - 2}{3} x - \frac{2}{9} \cdot 9 + 2$

Step 2.5.4

Multiply $2$ by $- 2$ .

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

Step 2.5.5

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

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

Step 2.5.6

Cancel the common factor of $9$ .

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Step 2.5.6.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

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

Step 2.5.6.2

Cancel the common factor.

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

Step 2.5.6.3

Rewrite the expression.

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

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

Step 2.6.2

Move the negative in front of the fraction.

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

Step 3

Combine the opposite terms in $- \frac{2 x^{2}}{9} - \frac{4 x}{3} - 2 + 2$ .

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Step 3.1

Add $- 2$ and $2$ .

$f (x) = - \frac{2 x^{2}}{9} - \frac{4 x}{3} + 0$

Step 3.2

Add $- \frac{2 x^{2}}{9} - \frac{4 x}{3}$ and $0$ .

$f (x) = - \frac{2 x^{2}}{9} - \frac{4 x}{3}$

Step 4

Reorder terms.

$f (x) = - (\frac{2}{9} x^{2}) - (\frac{4}{3} x)$

Step 5

Remove parentheses.

$f (x) = - \frac{2}{9} x^{2} - (\frac{4}{3} x)$

Step 6

Remove parentheses.

$f (x) = - \frac{2}{9} x^{2} - \frac{4}{3} x$