Algebra Examples

$3 - x - \frac{1}{2} x^{2}$

Step 1

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

$f (x) = 3 - x - \frac{x^{2}}{2}$

Step 2

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

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

Cancel the common factor.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.3.2.2

Cancel the common factor of $1$ .

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Step 3.3.2.2.1

Cancel the common factor.

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

Step 3.3.2.2.2

Rewrite the expression.

$x = - 1 \cdot 1$

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.

$f (- 1) = 3 - (- 1) - \frac{{(- 1)}^{2}}{2}$

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 $3$ as a fraction with denominator $1$ .

$f (- 1) = \frac{3}{1} - (- 1) - \frac{{(- 1)}^{2}}{2}$

Step 4.2.1.2

Multiply $\frac{3}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{3}{1} \cdot \frac{2}{2} - (- 1) - \frac{{(- 1)}^{2}}{2}$

Step 4.2.1.3

Multiply $\frac{3}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{3 \cdot 2}{2} - (- 1) - \frac{{(- 1)}^{2}}{2}$

Step 4.2.1.4

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

$f (- 1) = \frac{3 \cdot 2}{2} + \frac{- (- 1)}{1} - \frac{{(- 1)}^{2}}{2}$

Step 4.2.1.5

Multiply $\frac{- (- 1)}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{3 \cdot 2}{2} + \frac{- (- 1)}{1} \cdot \frac{2}{2} - \frac{{(- 1)}^{2}}{2}$

Step 4.2.1.6

Multiply $\frac{- (- 1)}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{3 \cdot 2}{2} + \frac{1 \cdot 2}{2} - \frac{{(- 1)}^{2}}{2}$

Step 4.2.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{3 \cdot 2 + 1 \cdot 2 - {(- 1)}^{2}}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply $3$ by $2$ .

$f (- 1) = \frac{6 + 1 \cdot 2 - {(- 1)}^{2}}{2}$

Step 4.2.3.2

Multiply $- (- 1) \cdot 2$ .

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Step 4.2.3.2.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = \frac{6 + 1 \cdot 2 - {(- 1)}^{2}}{2}$

Step 4.2.3.2.2

Multiply $2$ by $1$ .

$f (- 1) = \frac{6 + 2 - {(- 1)}^{2}}{2}$

Step 4.2.3.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.3.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 4.2.3.3.1.1

Raise $- 1$ to the power of $1$ .

$f (- 1) = \frac{6 + 2 + (- 1) {(- 1)}^{2}}{2}$

Step 4.2.3.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = \frac{6 + 2 + {(- 1)}^{1 + 2}}{2}$

Step 4.2.3.3.2

Add $1$ and $2$ .

$f (- 1) = \frac{6 + 2 + {(- 1)}^{3}}{2}$

Step 4.2.3.4

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{6 + 2 - 1}{2}$

Step 4.2.4

Simplify by adding and subtracting.

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Step 4.2.4.1

Add $6$ and $2$ .

$f (- 1) = \frac{8 - 1}{2}$

Step 4.2.4.2

Subtract $1$ from $8$ .

$f (- 1) = \frac{7}{2}$

Step 4.2.5

The final answer is $\frac{7}{2}$ .

$\frac{7}{2}$

Step 5

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

$(- 1, \frac{7}{2})$

Step 6