Algebra Examples

$a = 27 x - x^{2}$

Step 1

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 2

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

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Step 2.1

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

$x = - \frac{27}{2 (- 1)}$

Step 2.2

Remove parentheses.

$x = - \frac{27}{2 (- 1)}$

Step 2.3

Simplify $- \frac{27}{2 (- 1)}$ .

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Step 2.3.1

Multiply $2$ by $- 1$ .

$x = - \frac{27}{- 2}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{27}{2}$

Step 2.3.3

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

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{27}{2})$

Step 2.3.3.2

Multiply $\frac{27}{2}$ by $1$ .

$x = \frac{27}{2}$

Step 3

Evaluate $f (\frac{27}{2})$ .

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Step 3.1

Replace the variable $x$ with $\frac{27}{2}$ in the expression.

$f (\frac{27}{2}) = 27 (\frac{27}{2}) - {(\frac{27}{2})}^{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $27 (\frac{27}{2})$ .

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Step 3.2.1.1.1

Combine $27$ and $\frac{27}{2}$ .

$f (\frac{27}{2}) = \frac{27 \cdot 27}{2} - {(\frac{27}{2})}^{2}$

Step 3.2.1.1.2

Multiply $27$ by $27$ .

$f (\frac{27}{2}) = \frac{729}{2} - {(\frac{27}{2})}^{2}$

Step 3.2.1.2

Apply the product rule to $\frac{27}{2}$ .

$f (\frac{27}{2}) = \frac{729}{2} - \frac{27^{2}}{2^{2}}$

Step 3.2.1.3

Raise $27$ to the power of $2$ .

$f (\frac{27}{2}) = \frac{729}{2} - \frac{729}{2^{2}}$

Step 3.2.1.4

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

$f (\frac{27}{2}) = \frac{729}{2} - \frac{729}{4}$

Step 3.2.2

To write $\frac{729}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{27}{2}) = \frac{729}{2} \cdot \frac{2}{2} - \frac{729}{4}$

Step 3.2.3

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

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Step 3.2.3.1

Multiply $\frac{729}{2}$ by $\frac{2}{2}$ .

$f (\frac{27}{2}) = \frac{729 \cdot 2}{2 \cdot 2} - \frac{729}{4}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$f (\frac{27}{2}) = \frac{729 \cdot 2}{4} - \frac{729}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{27}{2}) = \frac{729 \cdot 2 - 729}{4}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $729$ by $2$ .

$f (\frac{27}{2}) = \frac{1458 - 729}{4}$

Step 3.2.5.2

Subtract $729$ from $1458$ .

$f (\frac{27}{2}) = \frac{729}{4}$

Step 3.2.6

The final answer is $\frac{729}{4}$ .

$\frac{729}{4}$

Step 4

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

$(\frac{27}{2}, \frac{729}{4})$

Step 5