Algebra Examples

$y = - x^{2} - x + 6$

Step 1

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 2

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

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Step 2.1

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

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

Step 2.2

Remove parentheses.

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

Step 2.3

Dividing two negative values results in a positive value.

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

Step 3

Evaluate $f (- \frac{1}{2})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 3.2.1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

Move ${(- 1)}^{2}$ .

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

Step 3.2.1.2.2

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

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Step 3.2.1.2.2.1

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

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

Step 3.2.1.2.2.2

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

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

Step 3.2.1.2.3

Add $2$ and $1$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

One to any power is one.

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

Step 3.2.1.5

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

$f (- \frac{1}{2}) = - \frac{1}{4} - (- \frac{1}{2}) + 6$

Step 3.2.1.6

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

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Step 3.2.1.6.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{1}{2}) = - \frac{1}{4} + 1 (\frac{1}{2}) + 6$

Step 3.2.1.6.2

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

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{1}{2} + 6$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

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

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2} + 6$

Step 3.2.2.2

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

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{2}{2 \cdot 2} + 6$

Step 3.2.2.3

Write $6$ as a fraction with denominator $1$ .

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{2}{2 \cdot 2} + \frac{6}{1}$

Step 3.2.2.4

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{2}{2 \cdot 2} + \frac{6}{1} \cdot \frac{4}{4}$

Step 3.2.2.5

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{2}{2 \cdot 2} + \frac{6 \cdot 4}{4}$

Step 3.2.2.6

Multiply $2$ by $2$ .

$f (- \frac{1}{2}) = - \frac{1}{4} + \frac{2}{4} + \frac{6 \cdot 4}{4}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{1}{2}) = \frac{- 1 + 2 + 6 \cdot 4}{4}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Multiply $6$ by $4$ .

$f (- \frac{1}{2}) = \frac{- 1 + 2 + 24}{4}$

Step 3.2.4.2

Add $- 1$ and $2$ .

$f (- \frac{1}{2}) = \frac{1 + 24}{4}$

Step 3.2.4.3

Add $1$ and $24$ .

$f (- \frac{1}{2}) = \frac{25}{4}$

Step 3.2.5

The final answer is $\frac{25}{4}$ .

$\frac{25}{4}$

Step 4

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

$(- \frac{1}{2}, \frac{25}{4})$

Step 5