Precalculus Examples

$f (x) = 2 x^{2} - 16 x - 6$

Step 1

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 2

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

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Step 2.1

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

$x = - \frac{- 16}{2 (2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 16}{2 (2)}$

Step 2.3

Simplify $- \frac{- 16}{2 (2)}$ .

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Step 2.3.1

Cancel the common factor of $- 16$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot - 8}{2 \cdot 2}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 2$ .

$x = - \frac{2 \cdot - 8}{2 (2)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 8}{2 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 8}{2}$

Step 2.3.2

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.2.1

Factor $2$ out of $- 8$ .

$x = - \frac{2 \cdot - 4}{2}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $2$ out of $2$ .

$x = - \frac{2 \cdot - 4}{2 (1)}$

Step 2.3.2.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 4}{2 \cdot 1}$

Step 2.3.2.2.3

Rewrite the expression.

$x = - \frac{- 4}{1}$

Step 2.3.2.2.4

Divide $- 4$ by $1$ .

$x = - - 4$

Step 2.3.3

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 3

Evaluate $f (4)$ .

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Step 3.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 {(4)}^{2} - 16 \cdot 4 - 6$

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

Raise $4$ to the power of $2$ .

$f (4) = 2 \cdot 16 - 16 \cdot 4 - 6$

Step 3.2.1.2

Multiply $2$ by $16$ .

$f (4) = 32 - 16 \cdot 4 - 6$

Step 3.2.1.3

Multiply $- 16$ by $4$ .

$f (4) = 32 - 64 - 6$

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $64$ from $32$ .

$f (4) = - 32 - 6$

Step 3.2.2.2

Subtract $6$ from $- 32$ .

$f (4) = - 38$

Step 3.2.3

The final answer is $- 38$ .

$- 38$

Step 4

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

$(4, - 38)$

Step 5