Precalculus Examples

$f (x) = - \frac{32}{{(20)}^{2}} x^{2} + x + 5$

Step 1

Simplify each term.

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Step 1.1

Raise $20$ to the power of $2$ .

$f (x) = - \frac{32}{400} x^{2} + x + 5$

Step 1.2

Cancel the common factor of $32$ and $400$ .

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Step 1.2.1

Factor $16$ out of $32$ .

$f (x) = - \frac{16 (2)}{400} x^{2} + x + 5$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $16$ out of $400$ .

$f (x) = - \frac{16 \cdot 2}{16 \cdot 25} x^{2} + x + 5$

Step 1.2.2.2

Cancel the common factor.

$f (x) = - \frac{16 \cdot 2}{16 \cdot 25} x^{2} + x + 5$

Step 1.2.2.3

Rewrite the expression.

$f (x) = - \frac{2}{25} x^{2} + x + 5$

Step 1.3

Combine $x^{2}$ and $\frac{2}{25}$ .

$f (x) = - \frac{x^{2} \cdot 2}{25} + x + 5$

Step 1.4

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

$f (x) = - \frac{2 x^{2}}{25} + x + 5$

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{1}{2 (- 0.08)}$

Step 3.2

Remove parentheses.

$x = - \frac{1}{2 (- 0.08)}$

Step 3.3

Simplify $- \frac{1}{2 (- 0.08)}$ .

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Step 3.3.1

Multiply $2$ by $- 0.08$ .

$x = - \frac{1}{- 0.16}$

Step 3.3.2

Divide $1$ by $- 0.16$ .

$x = - - 6.25$

Step 3.3.3

Multiply $- 1$ by $- 6.25$ .

$x = 6.25$

Step 4

Evaluate $f (6.25)$ .

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Step 4.1

Replace the variable $x$ with $6.25$ in the expression.

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + 6.25 + 5$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + 6.25 + 5$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $6.25$ as a fraction with denominator $1$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25}{1} + 5$

Step 4.2.2.2

Multiply $\frac{6.25}{1}$ by $\frac{25}{25}$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25}{1} \cdot \frac{25}{25} + 5$

Step 4.2.2.3

Multiply $\frac{6.25}{1}$ by $\frac{25}{25}$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25 \cdot 25}{25} + 5$

Step 4.2.2.4

Write $5$ as a fraction with denominator $1$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25 \cdot 25}{25} + \frac{5}{1}$

Step 4.2.2.5

Multiply $\frac{5}{1}$ by $\frac{25}{25}$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25 \cdot 25}{25} + \frac{5}{1} \cdot \frac{25}{25}$

Step 4.2.2.6

Multiply $\frac{5}{1}$ by $\frac{25}{25}$ .

$f (6.25) = - \frac{2 {(6.25)}^{2}}{25} + \frac{6.25 \cdot 25}{25} + \frac{5 \cdot 25}{25}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (6.25) = \frac{- 2 {(6.25)}^{2} + 6.25 \cdot 25 + 5 \cdot 25}{25}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Raise $6.25$ to the power of $2$ .

$f (6.25) = \frac{- 2 \cdot 39.0625 + 6.25 \cdot 25 + 5 \cdot 25}{25}$

Step 4.2.4.2

Multiply $- 2$ by $39.0625$ .

$f (6.25) = \frac{- 78.125 + 6.25 \cdot 25 + 5 \cdot 25}{25}$

Step 4.2.4.3

Multiply $6.25$ by $25$ .

$f (6.25) = \frac{- 78.125 + 156.25 + 5 \cdot 25}{25}$

Step 4.2.4.4

Multiply $5$ by $25$ .

$f (6.25) = \frac{- 78.125 + 156.25 + 125}{25}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Add $- 78.125$ and $156.25$ .

$f (6.25) = \frac{78.125 + 125}{25}$

Step 4.2.5.2

Add $78.125$ and $125$ .

$f (6.25) = \frac{203.125}{25}$

Step 4.2.5.3

Divide $203.125$ by $25$ .

$f (6.25) = 8.125$

Step 4.2.6

The final answer is $8.125$ .

$8.125$

Step 5

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

$(6.25, 8.125)$

Step 6