Precalculus Examples

$R (x) = - 25 x^{2} + 2200 x + 30000$

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 $b$ .

$x = - \frac{2200}{2 (- 25)}$

Step 2.2

Remove parentheses.

$x = - \frac{2200}{2 (- 25)}$

Step 2.3

Simplify $- \frac{2200}{2 (- 25)}$ .

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Step 2.3.1

Cancel the common factor of $2200$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $2200$ .

$x = - \frac{2 \cdot 1100}{2 \cdot - 25}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 25$ .

$x = - \frac{2 \cdot 1100}{2 (- 25)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 1100}{2 \cdot - 25}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{1100}{- 25}$

Step 2.3.2

Cancel the common factor of $1100$ and $- 25$ .

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Step 2.3.2.1

Factor $25$ out of $1100$ .

$x = - \frac{25 (44)}{- 25}$

Step 2.3.2.2

Move the negative one from the denominator of $\frac{44}{- 1}$ .

$x = - (- 1 \cdot 44)$

Step 2.3.3

Multiply $- (- 1 \cdot 44)$ .

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Step 2.3.3.1

Multiply $- 1$ by $44$ .

$x = - - 44$

Step 2.3.3.2

Multiply $- 1$ by $- 44$ .

$x = 44$

Step 3

Evaluate $f (44)$ .

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Step 3.1

Replace the variable $x$ with $44$ in the expression.

$R (44) = - 25 {(44)}^{2} + 2200 (44) + 30000$

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

Raise $44$ to the power of $2$ .

$R (44) = - 25 \cdot 1936 + 2200 (44) + 30000$

Step 3.2.1.2

Multiply $- 25$ by $1936$ .

$R (44) = - 48400 + 2200 (44) + 30000$

Step 3.2.1.3

Multiply $2200$ by $44$ .

$R (44) = - 48400 + 96800 + 30000$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 48400$ and $96800$ .

$R (44) = 48400 + 30000$

Step 3.2.2.2

Add $48400$ and $30000$ .

$R (44) = 78400$

Step 3.2.3

The final answer is $78400$ .

$78400$

Step 4

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

$(44, 78400)$

Step 5