Algebra Examples

$f (x) = - x^{2} - 2 x - 6$

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 $b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

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

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Step 2.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 2$ .

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

Step 2.3.1.2

Move the negative one from the denominator of $\frac{- 1}{- 1}$ .

$x = - (- 1 \cdot - 1)$

Step 2.3.2

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$x = - - - 1$

Step 2.3.3

Multiply $- - - 1$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = - 1 \cdot 1$

Step 2.3.3.2

Multiply $- 1$ by $1$ .

$x = - 1$

Step 3

Evaluate $f (- 1)$ .

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Step 3.1

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

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

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

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

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

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

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

Step 3.2.1.1.1.2

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

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

Step 3.2.1.1.2

Add $1$ and $2$ .

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

Step 3.2.1.2

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

$f (- 1) = - 1 - 2 \cdot - 1 - 6$

Step 3.2.1.3

Multiply $- 2$ by $- 1$ .

$f (- 1) = - 1 + 2 - 6$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 1$ and $2$ .

$f (- 1) = 1 - 6$

Step 3.2.2.2

Subtract $6$ from $1$ .

$f (- 1) = - 5$

Step 3.2.3

The final answer is $- 5$ .

$- 5$

Step 4

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

$(- 1, - 5)$

Step 5