Algebra Examples

$y = \frac{1}{400} x^{2} - x + 150$

Step 1

Combine $\frac{1}{400}$ and $x^{2}$ .

$f (x) = \frac{x^{2}}{400} - x + 150$

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{- 1}{2 (0.0025)}$

Step 3.2

Remove parentheses.

$x = - \frac{- 1}{2 (0.0025)}$

Step 3.3

Simplify $- \frac{- 1}{2 (0.0025)}$ .

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Step 3.3.1

Multiply $2$ by $0.0025$ .

$x = - \frac{- 1}{0.005}$

Step 3.3.2

Divide $- 1$ by $0.005$ .

$x = - - 200$

Step 3.3.3

Multiply $- 1$ by $- 200$ .

$x = 200$

Step 4

Evaluate $f (200)$ .

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Step 4.1

Replace the variable $x$ with $200$ in the expression.

$f (200) = \frac{{(200)}^{2}}{400} - (200) + 150$

Step 4.2

Simplify the result.

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Step 4.2.1

Find the common denominator.

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Step 4.2.1.1

Write $- (200)$ as a fraction with denominator $1$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- (200)}{1} + 150$

Step 4.2.1.2

Multiply $\frac{- (200)}{1}$ by $\frac{400}{400}$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- (200)}{1} \cdot \frac{400}{400} + 150$

Step 4.2.1.3

Multiply $\frac{- (200)}{1}$ by $\frac{400}{400}$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- 200 \cdot 400}{400} + 150$

Step 4.2.1.4

Write $150$ as a fraction with denominator $1$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- 200 \cdot 400}{400} + \frac{150}{1}$

Step 4.2.1.5

Multiply $\frac{150}{1}$ by $\frac{400}{400}$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- 200 \cdot 400}{400} + \frac{150}{1} \cdot \frac{400}{400}$

Step 4.2.1.6

Multiply $\frac{150}{1}$ by $\frac{400}{400}$ .

$f (200) = \frac{{(200)}^{2}}{400} + \frac{- 200 \cdot 400}{400} + \frac{150 \cdot 400}{400}$

Step 4.2.2

Combine the numerators over the common denominator.

$f (200) = \frac{{(200)}^{2} - 200 \cdot 400 + 150 \cdot 400}{400}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Raise $200$ to the power of $2$ .

$f (200) = \frac{40000 - 200 \cdot 400 + 150 \cdot 400}{400}$

Step 4.2.3.2

Multiply $- (200) \cdot 400$ .

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Step 4.2.3.2.1

Multiply $- 1$ by $200$ .

$f (200) = \frac{40000 - 200 \cdot 400 + 150 \cdot 400}{400}$

Step 4.2.3.2.2

Multiply $- 200$ by $400$ .

$f (200) = \frac{40000 - 80000 + 150 \cdot 400}{400}$

Step 4.2.3.3

Multiply $150$ by $400$ .

$f (200) = \frac{40000 - 80000 + 60000}{400}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Subtract $80000$ from $40000$ .

$f (200) = \frac{- 40000 + 60000}{400}$

Step 4.2.4.2

Add $- 40000$ and $60000$ .

$f (200) = \frac{20000}{400}$

Step 4.2.4.3

Divide $20000$ by $400$ .

$f (200) = 50$

Step 4.2.5

The final answer is $50$ .

$50$

Step 5

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

$(200, 50)$

Step 6