Algebra Examples

$c = 400 - 5 x + 0.125 x^{2}$

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

Tap for more steps...

Step 2.1

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

$x = - \frac{- 5}{2 (0.125)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 5}{2 (0.125)}$

Step 2.3

Simplify $- \frac{- 5}{2 (0.125)}$ .

Tap for more steps...

Step 2.3.1

Multiply $2$ by $0.125$ .

$x = - \frac{- 5}{0.25}$

Step 2.3.2

Divide $- 5$ by $0.25$ .

$x = - - 20$

Step 2.3.3

Multiply $- 1$ by $- 20$ .

$x = 20$

Step 3

Evaluate $f (20)$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $20$ in the expression.

$f (20) = 400 - 5 \cdot 20 + 0.125 {(20)}^{2}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $- 5$ by $20$ .

$f (20) = 400 - 100 + 0.125 {(20)}^{2}$

Step 3.2.1.2

Raise $20$ to the power of $2$ .

$f (20) = 400 - 100 + 0.125 \cdot 400$

Step 3.2.1.3

Multiply $0.125$ by $400$ .

$f (20) = 400 - 100 + 50$

Step 3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.2.2.1

Subtract $100$ from $400$ .

$f (20) = 300 + 50$

Step 3.2.2.2

Add $300$ and $50$ .

$f (20) = 350$

Step 3.2.3

The final answer is $350$ .

$350$

Step 4

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

$(20, 350)$

Step 5