Finite Math Examples

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

Find the x-value the minimum occurs at. Use this value to find the minimum value.

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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 $b$ .

$x = - \frac{5}{2 (2)}$

Remove the extra parentheses from the expression $- \frac{5}{2 (2)}$ .

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Remove the parentheses from the numerator.

$x = - \frac{5}{2 (2)}$

Remove the parentheses from the denominator.

$x = - \frac{5}{2 (2)}$

Multiply $2$ by $2$ .

$x = - \frac{5}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = - \frac{5}{4}$

Replace the variable $x$ with $- \frac{5}{4}$ in the expression.

$f (- \frac{5}{4}) = 2 {(- \frac{5}{4})}^{2} + 5 (- \frac{5}{4}) - 6$

Simplify the result.

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Simplify each term.

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Apply the product rule to $- \frac{5}{4}$ .

$f (- \frac{5}{4}) = 2 ({(- 1)}^{2} {(\frac{5}{4})}^{2}) + 5 (- \frac{5}{4}) - 6$

Raise $- 1$ to the power of $2$ .

$f (- \frac{5}{4}) = 2 (1 {(\frac{5}{4})}^{2}) + 5 (- \frac{5}{4}) - 6$

Multiply ${(\frac{5}{4})}^{2}$ by $1$ .

$f (- \frac{5}{4}) = 2 {(\frac{5}{4})}^{2} + 5 (- \frac{5}{4}) - 6$

Apply the product rule to $\frac{5}{4}$ .

$f (- \frac{5}{4}) = 2 (\frac{5^{2}}{4^{2}}) + 5 (- \frac{5}{4}) - 6$

Raise $5$ to the power of $2$ .

$f (- \frac{5}{4}) = 2 (\frac{25}{4^{2}}) + 5 (- \frac{5}{4}) - 6$

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

$f (- \frac{5}{4}) = 2 (\frac{25}{16}) + 5 (- \frac{5}{4}) - 6$

Cancel the common factor of $2$ .

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Write $2$ as a fraction with denominator $1$ .

$f (- \frac{5}{4}) = \frac{2}{1} \cdot \frac{25}{16} + 5 (- \frac{5}{4}) - 6$

Factor out the greatest common factor $2$ .

$f (- \frac{5}{4}) = \frac{2 \cdot 1}{1} \cdot \frac{25}{2 \cdot 8} + 5 (- \frac{5}{4}) - 6$

Cancel the common factor.

$f (- \frac{5}{4}) = \frac{2 \cdot 1}{1} \cdot \frac{25}{2 \cdot 8} + 5 (- \frac{5}{4}) - 6$

Rewrite the expression.

$f (- \frac{5}{4}) = \frac{1}{1} \cdot \frac{25}{8} + 5 (- \frac{5}{4}) - 6$

Multiply $\frac{1}{1}$ and $\frac{25}{8}$ .

$f (- \frac{5}{4}) = \frac{25}{8} + 5 (- \frac{5}{4}) - 6$

Simplify $5 (- \frac{5}{4})$ .

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Multiply $- 1$ by $5$ .

$f (- \frac{5}{4}) = \frac{25}{8} - 5 (\frac{5}{4}) - 6$

Write $- 5$ as a fraction with denominator $1$ .

$f (- \frac{5}{4}) = \frac{25}{8} + \frac{- 5}{1} \cdot \frac{5}{4} - 6$

Multiply $\frac{- 5}{1}$ and $\frac{5}{4}$ .

$f (- \frac{5}{4}) = \frac{25}{8} + \frac{- 5 \cdot 5}{4} - 6$

Multiply $- 5$ by $5$ .

$f (- \frac{5}{4}) = \frac{25}{8} + \frac{- 25}{4} - 6$

Move the negative in front of the fraction.

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25}{4} - 6$

To write $\frac{25}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25}{4} \cdot \frac{2}{2} - 6$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{4 \cdot 2} - 6$

Multiply $4$ by $2$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{8} - 6$

Combine the numerators over the common denominator.

$f (- \frac{5}{4}) = \frac{25 - (25 \cdot 2)}{8} - 6$

Simplify the numerator.

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Multiply $25$ by $2$ .

$f (- \frac{5}{4}) = \frac{25 - 1 \cdot 50}{8} - 6$

Multiply $- 1$ by $50$ .

$f (- \frac{5}{4}) = \frac{25 - 50}{8} - 6$

Subtract $50$ from $25$ .

$f (- \frac{5}{4}) = \frac{- 25}{8} - 6$

Move the negative in front of the fraction.

$f (- \frac{5}{4}) = - \frac{25}{8} - 6$

To write $- \frac{- 6}{1}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{5}{4}) = - \frac{25}{8} + \frac{- 6}{1} \cdot \frac{8}{8}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$f (- \frac{5}{4}) = - \frac{25}{8} + \frac{- 6 \cdot 8}{1 \cdot 8}$

Multiply $8$ by $1$ .

$f (- \frac{5}{4}) = - \frac{25}{8} + \frac{- 6 \cdot 8}{8}$

Combine the numerators over the common denominator.

$f (- \frac{5}{4}) = \frac{- 1 \cdot 25 - 6 \cdot 8}{8}$

Simplify the numerator.

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Multiply $- 1$ by $25$ .

$f (- \frac{5}{4}) = \frac{- 25 - 6 \cdot 8}{8}$

Multiply $- 6$ by $8$ .

$f (- \frac{5}{4}) = \frac{- 25 - 48}{8}$

Subtract $48$ from $- 25$ .

$f (- \frac{5}{4}) = \frac{- 73}{8}$

Move the negative in front of the fraction.

$f (- \frac{5}{4}) = - \frac{73}{8}$

The final answer is $- \frac{73}{8}$ .

$- \frac{73}{8}$

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