Find the Maximum/Minimum Value p(x)=-5x^2+240x-2560

$p (x) = - 5 x^{2} + 240 x - 2560$

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 $x = - \frac{b}{2 a}$ .

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

$x = - \frac{240}{2 (- 5)}$

Remove parentheses.

$x = - \frac{240}{2 (- 5)}$

Simplify $- \frac{240}{2 (- 5)}$ .

Cancel the common factor of $240$ and $2$ .

Factor $2$ out of $240$ .

$x = - \frac{2 \cdot 120}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$x = - \frac{2 \cdot 120}{2 (- 5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 120}{2 \cdot - 5}$

Rewrite the expression.

$x = - \frac{120}{- 5}$

Cancel the common factor of $120$ and $- 5$ .

Factor $5$ out of $120$ .

$x = - \frac{5 (24)}{- 5}$

Move the negative one from the denominator of $\frac{24}{- 1}$ .

$x = - (- 1 \cdot 24)$

Multiply $- (- 1 \cdot 24)$ .

Multiply $- 1$ by $24$ .

$x = - - 24$

Multiply $- 1$ by $- 24$ .

$x = 24$

Step 3

Evaluate $f (24)$ .

Replace the variable $x$ with $24$ in the expression.

$p (24) = - 5 {(24)}^{2} + 240 (24) - 2560$

Simplify the result.

Simplify each term.

Raise $24$ to the power of $2$ .

$p (24) = - 5 \cdot 576 + 240 (24) - 2560$

Multiply $- 5$ by $576$ .

$p (24) = - 2880 + 240 (24) - 2560$

Multiply $240$ by $24$ .

$p (24) = - 2880 + 5760 - 2560$

Simplify by adding and subtracting.

Add $- 2880$ and $5760$ .

$p (24) = 2880 - 2560$

Subtract $2560$ from $2880$ .

$p (24) = 320$

The final answer is $320$ .

$320$

Step 4

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

$(24, 320)$

Step 5

Find the Maximum/Minimum Value p(x)=-5x^2+240x-2560

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