Finite Math Examples

$P (x) = 110 (55) - x^{2} - 1000$

Step 1

Multiply $110$ by $55$ .

$P (x) = 6050 - x^{2} - 1000$

Step 2

Subtract $1000$ from $6050$ .

$P (x) = - x^{2} + 5050$

Step 3

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 4

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

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Step 4.1

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

$x = - \frac{0}{2 (- 1)}$

Step 4.2

Remove parentheses.

$x = - \frac{0}{2 (- 1)}$

Step 4.3

Simplify $- \frac{0}{2 (- 1)}$ .

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Step 4.3.1

Cancel the common factor of $0$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $0$ .

$x = - \frac{2 (0)}{2 (- 1)}$

Step 4.3.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$x = - (- 1 \cdot 0)$

Step 4.3.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$x = - - 0$

Step 4.3.3

Multiply $- - 0$ .

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Step 4.3.3.1

Multiply $- 1$ by $0$ .

$x = - 0$

Step 4.3.3.2

Multiply $- 1$ by $0$ .

$x = 0$

Step 5

Evaluate $f (0)$ .

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$P (0) = - {(0)}^{2} + 5050$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$P (0) = - 0 + 5050$

Step 5.2.1.2

Multiply $- 1$ by $0$ .

$P (0) = 0 + 5050$

Step 5.2.2

Add $0$ and $5050$ .

$P (0) = 5050$

Step 5.2.3

The final answer is $5050$ .

$5050$

Step 6

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

$(0, 5050)$

Step 7