Finite Math Examples

$f (x) = x^{2} - 4 x + 2$

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

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

$x = - \frac{b}{2 a}$

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

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

Remove the extra parentheses from the expression $- \frac{- 4}{2 (1)}$ .

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Remove the parentheses from the numerator.

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

Remove the parentheses from the denominator.

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factors.

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Factor $2$ out of $2 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$x = 2$

Replace the variable $x$ with $2$ in the expression.

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

Simplify the result.

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Simplify each term.

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

$f (2) = 4 - 4 \cdot 2 + 2$

Multiply $- 4$ by $2$ .

$f (2) = 4 - 8 + 2$

Simplify by adding and subtracting.

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Subtract $8$ from $4$ .

$f (2) = - 4 + 2$

Add $- 4$ and $2$ .

$f (2) = - 2$

The final answer is $- 2$ .

$- 2$

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

$(2, - 2)$

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