Algebra Examples

$y = - 3 x (x - 3)$

Step 1

Apply the distributive property.

$f (x) = - 3 x \cdot x - 3 x \cdot - 3$

Step 2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1

Move $x$ .

$f (x) = - 3 (x \cdot x) - 3 x \cdot - 3$

Step 2.2

Multiply $x$ by $x$ .

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

Step 3

Multiply $- 3$ by $- 3$ .

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

Step 4

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 5

Find the value of $x = - \frac{b}{2 a}$ .

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Step 5.1

Substitute in the values of $a$ and $b$ .

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

Step 5.2

Remove parentheses.

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

Step 5.3

Simplify $- \frac{9}{2 (- 3)}$ .

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Step 5.3.1

Cancel the common factor of $9$ and $- 3$ .

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Step 5.3.1.1

Factor $3$ out of $9$ .

$x = - \frac{3 (3)}{2 (- 3)}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $2 (- 3)$ .

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

Step 5.3.1.2.2

Cancel the common factor.

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

Step 5.3.1.2.3

Rewrite the expression.

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

Step 5.3.2

Multiply $2$ by $- 1$ .

$x = - \frac{3}{- 2}$

Step 5.3.3

Move the negative in front of the fraction.

$x = - - \frac{3}{2}$

Step 5.3.4

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

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Step 5.3.4.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{3}{2})$

Step 5.3.4.2

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

$x = \frac{3}{2}$

Step 6

Evaluate $f (\frac{3}{2})$ .

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Step 6.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the product rule to $\frac{3}{2}$ .

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

Step 6.2.1.2

Raise $3$ to the power of $2$ .

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

Step 6.2.1.3

Raise $2$ to the power of $2$ .

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

Step 6.2.1.4

Multiply $- 3 (\frac{9}{4})$ .

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Step 6.2.1.4.1

Combine $- 3$ and $\frac{9}{4}$ .

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

Step 6.2.1.4.2

Multiply $- 3$ by $9$ .

$f (\frac{3}{2}) = \frac{- 27}{4} + 9 (\frac{3}{2})$

Step 6.2.1.5

Move the negative in front of the fraction.

$f (\frac{3}{2}) = - \frac{27}{4} + 9 (\frac{3}{2})$

Step 6.2.1.6

Multiply $9 (\frac{3}{2})$ .

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Step 6.2.1.6.1

Combine $9$ and $\frac{3}{2}$ .

$f (\frac{3}{2}) = - \frac{27}{4} + \frac{9 \cdot 3}{2}$

Step 6.2.1.6.2

Multiply $9$ by $3$ .

$f (\frac{3}{2}) = - \frac{27}{4} + \frac{27}{2}$

Step 6.2.2

To write $\frac{27}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{3}{2}) = - \frac{27}{4} + \frac{27}{2} \cdot \frac{2}{2}$

Step 6.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

Multiply $\frac{27}{2}$ by $\frac{2}{2}$ .

$f (\frac{3}{2}) = - \frac{27}{4} + \frac{27 \cdot 2}{2 \cdot 2}$

Step 6.2.3.2

Multiply $2$ by $2$ .

$f (\frac{3}{2}) = - \frac{27}{4} + \frac{27 \cdot 2}{4}$

Step 6.2.4

Combine the numerators over the common denominator.

$f (\frac{3}{2}) = \frac{- 27 + 27 \cdot 2}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $27$ by $2$ .

$f (\frac{3}{2}) = \frac{- 27 + 54}{4}$

Step 6.2.5.2

Add $- 27$ and $54$ .

$f (\frac{3}{2}) = \frac{27}{4}$

Step 6.2.6

The final answer is $\frac{27}{4}$ .

$\frac{27}{4}$

Step 7

Use the $x$ and $y$ values to find where the maximum occurs.

$(\frac{3}{2}, \frac{27}{4})$

Step 8