Find the Maximum/Minimum Value f(x)=-2x^2+20x-51

$f (x) = - 2 x^{2} + 20 x - 51$

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 $b$ .

$x = - \frac{20}{2 (- 2)}$

Remove parentheses.

$x = - \frac{20}{2 (- 2)}$

Simplify $- \frac{20}{2 (- 2)}$ .

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$x = - \frac{2 \cdot 10}{2 \cdot - 2}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 10}{2 (- 2)}$

Cancel the common factor.

$x = - \frac{2 \cdot 10}{2 \cdot - 2}$

Rewrite the expression.

$x = - \frac{10}{- 2}$

Cancel the common factor of $10$ and $- 2$ .

Factor $2$ out of $10$ .

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

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$x = - (- 1 \cdot 5)$

Multiply $- (- 1 \cdot 5)$ .

Multiply $- 1$ by $5$ .

$x = - - 5$

Multiply $- 1$ by $- 5$ .

$x = 5$

Step 3

Evaluate $f (5)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 2 {(5)}^{2} + 20 (5) - 51$

Simplify the result.

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = - 2 \cdot 25 + 20 (5) - 51$

Multiply $- 2$ by $25$ .

$f (5) = - 50 + 20 (5) - 51$

Multiply $20$ by $5$ .

$f (5) = - 50 + 100 - 51$

Simplify by adding and subtracting.

Add $- 50$ and $100$ .

$f (5) = 50 - 51$

Subtract $51$ from $50$ .

$f (5) = - 1$

The final answer is $- 1$ .

$- 1$

Step 4

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

$(5, - 1)$

Step 5

Find the Maximum/Minimum Value f(x)=-2x^2+20x-51

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