Find the Maximum/Minimum Value 100x^2-1800x

$100 x^{2} - 1800 x$

Step 1

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{- 1800}{2 (100)}$

Remove parentheses.

$x = - \frac{- 1800}{2 (100)}$

Simplify $- \frac{- 1800}{2 (100)}$ .

Cancel the common factor of $- 1800$ and $2$ .

Factor $2$ out of $- 1800$ .

$x = - \frac{2 \cdot - 900}{2 \cdot 100}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 100$ .

$x = - \frac{2 \cdot - 900}{2 (100)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 900}{2 \cdot 100}$

Rewrite the expression.

$x = - \frac{- 900}{100}$

Cancel the common factor of $- 900$ and $100$ .

Factor $100$ out of $- 900$ .

$x = - \frac{100 \cdot - 9}{100}$

Cancel the common factors.

Factor $100$ out of $100$ .

$x = - \frac{100 \cdot - 9}{100 (1)}$

Cancel the common factor.

$x = - \frac{100 \cdot - 9}{100 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 9}{1}$

Divide $- 9$ by $1$ .

$x = - - 9$

Multiply $- 1$ by $- 9$ .

$x = 9$

Step 3

Evaluate $f (9)$ .

Replace the variable $x$ with $9$ in the expression.

$f (9) = 100 {(9)}^{2} - 1800 \cdot 9$

Simplify the result.

Simplify each term.

Raise $9$ to the power of $2$ .

$f (9) = 100 \cdot 81 - 1800 \cdot 9$

Multiply $100$ by $81$ .

$f (9) = 8100 - 1800 \cdot 9$

Multiply $- 1800$ by $9$ .

$f (9) = 8100 - 16200$

Subtract $16200$ from $8100$ .

$f (9) = - 8100$

The final answer is $- 8100$ .

$- 8100$

Step 4

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

$(9, - 8100)$

Step 5

Find the Maximum/Minimum Value 100x^2-1800x

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