Algebra Examples

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

Step 1

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

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

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 $b$ .

$x = - \frac{- 1}{2 (0.5)}$

Step 3.2

Remove parentheses.

$x = - \frac{- 1}{2 (0.5)}$

Step 3.3

Simplify $- \frac{- 1}{2 (0.5)}$ .

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Step 3.3.1

Multiply $2$ by $0.5$ .

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

Step 3.3.2

Divide $- 1$ by $1$ .

$x = - - 1$

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 4

Evaluate $f (1)$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{{(1)}^{2}}{2} - (1)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$f (1) = \frac{1}{2} - (1)$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = \frac{1}{2} - 1$

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (1) = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 4.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$f (1) = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{1 - 1 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 1$ by $2$ .

$f (1) = \frac{1 - 2}{2}$

Step 4.2.5.2

Subtract $2$ from $1$ .

$f (1) = \frac{- 1}{2}$

Step 4.2.6

Move the negative in front of the fraction.

$f (1) = - \frac{1}{2}$

Step 4.2.7

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 5

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

$(1, - \frac{1}{2})$

Step 6