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Examples

$f (x) = x^{2} - 6 x + 5$

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

$b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Simplify

$- \frac{- 6}{2 (1)}$ .

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Step 2.3.1

Cancel the common factor of

$- 6$ and

$2$ .

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Step 2.3.1.1

Factor

$2$ out of

$- 6$ .

$x = - \frac{2 \cdot - 3}{2 \cdot 1}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor

$2$ out of

$2 \cdot 1$ .

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

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 3}{2 \cdot 1}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 3}{1}$

Step 2.3.1.2.4

Divide

$- 3$ by

$1$ .

$x = - - 3$

$x = - - 3$

$x = - - 3$

Step 2.3.2

Multiply

$- 1$ by

$- 3$ .

$x = 3$

$x = 3$

$x = 3$

Step 3

Evaluate

$f (3)$ .

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Step 3.1

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = {(3)}^{2} - 6 \cdot 3 + 5$

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

$3$ to the power of

$2$ .

$f (3) = 9 - 6 \cdot 3 + 5$

Step 3.2.1.2

Multiply

$- 6$ by

$3$ .

$f (3) = 9 - 18 + 5$

$f (3) = 9 - 18 + 5$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract

$18$ from

$9$ .

$f (3) = - 9 + 5$

Step 3.2.2.2

Add

$- 9$ and

$5$ .

$f (3) = - 4$

$f (3) = - 4$

Step 3.2.3

The final answer is

$- 4$ .

$- 4$

$- 4$

$- 4$

Step 4

Use the

$x$ and

$y$ values to find where the minimum occurs.

$(3, - 4)$

Step 5

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$