Algebra Examples

$g (x) = \frac{1}{20} x (x - 100)$

Step 1

Apply the distributive property.

$g (x) = \frac{1}{20} \cdot (x \cdot x + x \cdot - 100)$

Step 2

Simplify the expression.

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Step 2.1

Multiply $x$ by $x$ .

$g (x) = \frac{1}{20} \cdot (x^{2} + x \cdot - 100)$

Step 2.2

Move $- 100$ to the left of $x$ .

$g (x) = \frac{1}{20} \cdot (x^{2} - 100 \cdot x)$

Step 3

Apply the distributive property.

$g (x) = \frac{1}{20} x^{2} + \frac{1}{20} \cdot (- 100 x)$

Step 4

Combine $\frac{1}{20}$ and $x^{2}$ .

$g (x) = \frac{x^{2}}{20} + \frac{1}{20} \cdot (- 100 x)$

Step 5

Cancel the common factor of $20$ .

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Step 5.1

Factor $20$ out of $- 100 x$ .

$g (x) = \frac{x^{2}}{20} + \frac{1}{20} \cdot (20 (- 5 x))$

Step 5.2

Cancel the common factor.

$g (x) = \frac{x^{2}}{20} + \frac{1}{20} \cdot (20 (- 5 x))$

Step 5.3

Rewrite the expression.

$g (x) = \frac{x^{2}}{20} - 5 x$

Step 6

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 7

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

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Step 7.1

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

$x = - \frac{- 5}{2 (0.05)}$

Step 7.2

Remove parentheses.

$x = - \frac{- 5}{2 (0.05)}$

Step 7.3

Simplify $- \frac{- 5}{2 (0.05)}$ .

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Step 7.3.1

Multiply $2$ by $0.05$ .

$x = - \frac{- 5}{0.1}$

Step 7.3.2

Divide $- 5$ by $0.1$ .

$x = - - 50$

Step 7.3.3

Multiply $- 1$ by $- 50$ .

$x = 50$

Step 8

Evaluate $f (50)$ .

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Step 8.1

Replace the variable $x$ with $50$ in the expression.

$g (50) = \frac{{(50)}^{2}}{20} - 5 \cdot 50$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $50$ to the power of $2$ .

$g (50) = \frac{2500}{20} - 5 \cdot 50$

Step 8.2.1.2

Divide $2500$ by $20$ .

$g (50) = 125 - 5 \cdot 50$

Step 8.2.1.3

Multiply $- 5$ by $50$ .

$g (50) = 125 - 250$

Step 8.2.2

Subtract $250$ from $125$ .

$g (50) = - 125$

Step 8.2.3

The final answer is $- 125$ .

$- 125$

Step 9

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

$(50, - 125)$

Step 10