Find the Maximum/Minimum Value -2x^2+145x

$- 2 x^{2} + 145 x$

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

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

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

$x = - \frac{145}{2 (- 2)}$

Remove parentheses.

$x = - \frac{145}{2 (- 2)}$

Simplify $- \frac{145}{2 (- 2)}$ .

Multiply $2$ by $- 2$ .

$x = - \frac{145}{- 4}$

Move the negative in front of the fraction.

$x = - - \frac{145}{4}$

Multiply $- - \frac{145}{4}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{145}{4})$

Multiply $\frac{145}{4}$ by $1$ .

$x = \frac{145}{4}$

Evaluate $f (\frac{145}{4})$ .

Replace the variable $x$ with $\frac{145}{4}$ in the expression.

$f (\frac{145}{4}) = - 2 {(\frac{145}{4})}^{2} + 145 (\frac{145}{4})$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{145}{4}$ .

$f (\frac{145}{4}) = - 2 \frac{145^{2}}{4^{2}} + 145 (\frac{145}{4})$

Raise $145$ to the power of $2$ .

$f (\frac{145}{4}) = - 2 \frac{21025}{4^{2}} + 145 (\frac{145}{4})$

Raise $4$ to the power of $2$ .

$f (\frac{145}{4}) = - 2 (\frac{21025}{16}) + 145 (\frac{145}{4})$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$f (\frac{145}{4}) = 2 (- 1) (\frac{21025}{16}) + 145 (\frac{145}{4})$

Factor $2$ out of $16$ .

$f (\frac{145}{4}) = 2 \cdot (- 1 \frac{21025}{2 \cdot 8}) + 145 (\frac{145}{4})$

Cancel the common factor.

$f (\frac{145}{4}) = 2 \cdot (- 1 \frac{21025}{2 \cdot 8}) + 145 (\frac{145}{4})$

Rewrite the expression.

$f (\frac{145}{4}) = - 1 (\frac{21025}{8}) + 145 (\frac{145}{4})$

Rewrite $- 1 (\frac{21025}{8})$ as $- (\frac{21025}{8})$ .

$f (\frac{145}{4}) = - (\frac{21025}{8}) + 145 (\frac{145}{4})$

Multiply $145 (\frac{145}{4})$ .

Combine $145$ and $\frac{145}{4}$ .

$f (\frac{145}{4}) = - \frac{21025}{8} + \frac{145 \cdot 145}{4}$

Multiply $145$ by $145$ .

$f (\frac{145}{4}) = - \frac{21025}{8} + \frac{21025}{4}$

To write $\frac{21025}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{145}{4}) = - \frac{21025}{8} + \frac{21025}{4} \cdot \frac{2}{2}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{21025}{4}$ by $\frac{2}{2}$ .

$f (\frac{145}{4}) = - \frac{21025}{8} + \frac{21025 \cdot 2}{4 \cdot 2}$

Multiply $4$ by $2$ .

$f (\frac{145}{4}) = - \frac{21025}{8} + \frac{21025 \cdot 2}{8}$

Combine the numerators over the common denominator.

$f (\frac{145}{4}) = \frac{- 21025 + 21025 \cdot 2}{8}$

Simplify the numerator.

Multiply $21025$ by $2$ .

$f (\frac{145}{4}) = \frac{- 21025 + 42050}{8}$

Add $- 21025$ and $42050$ .

$f (\frac{145}{4}) = \frac{21025}{8}$

The final answer is $\frac{21025}{8}$ .

$\frac{21025}{8}$

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

$(\frac{145}{4}, \frac{21025}{8})$

Find the Maximum/Minimum Value -2x^2+145x

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