Finite Math Examples

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

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 $b$ .

$x = - \frac{5}{2 (4)}$

Step 2.2

Remove parentheses.

$x = - \frac{5}{2 (4)}$

Step 2.3

Multiply $2$ by $4$ .

$x = - \frac{5}{8}$

Step 3

Evaluate $f (- \frac{5}{8})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{5}{8}$ in the expression.

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

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to $- \frac{5}{8}$ .

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

Step 3.2.1.1.2

Apply the product rule to $\frac{5}{8}$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply $\frac{5^{2}}{8^{2}}$ by $1$ .

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

Step 3.2.1.4

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

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

Step 3.2.1.5

Raise $8$ to the power of $2$ .

$f (- \frac{5}{8}) = 4 (\frac{25}{64}) + 5 (- \frac{5}{8}) - 6$

Step 3.2.1.6

Cancel the common factor of $4$ .

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Step 3.2.1.6.1

Factor $4$ out of $64$ .

$f (- \frac{5}{8}) = 4 (\frac{25}{4 (16)}) + 5 (- \frac{5}{8}) - 6$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{5}{8}) = 4 (\frac{25}{4 \cdot 16}) + 5 (- \frac{5}{8}) - 6$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{5}{8}) = \frac{25}{16} + 5 (- \frac{5}{8}) - 6$

Step 3.2.1.7

Multiply $5 (- \frac{5}{8})$ .

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Step 3.2.1.7.1

Multiply $- 1$ by $5$ .

$f (- \frac{5}{8}) = \frac{25}{16} - 5 (\frac{5}{8}) - 6$

Step 3.2.1.7.2

Combine $- 5$ and $\frac{5}{8}$ .

$f (- \frac{5}{8}) = \frac{25}{16} + \frac{- 5 \cdot 5}{8} - 6$

Step 3.2.1.7.3

Multiply $- 5$ by $5$ .

$f (- \frac{5}{8}) = \frac{25}{16} + \frac{- 25}{8} - 6$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25}{8} - 6$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{25}{8}$ by $\frac{2}{2}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - (\frac{25}{8} \cdot \frac{2}{2}) - 6$

Step 3.2.2.2

Multiply $\frac{25}{8}$ by $\frac{2}{2}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{8 \cdot 2} - 6$

Step 3.2.2.3

Write $- 6$ as a fraction with denominator $1$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{8 \cdot 2} + \frac{- 6}{1}$

Step 3.2.2.4

Multiply $\frac{- 6}{1}$ by $\frac{16}{16}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{8 \cdot 2} + \frac{- 6}{1} \cdot \frac{16}{16}$

Step 3.2.2.5

Multiply $\frac{- 6}{1}$ by $\frac{16}{16}$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{8 \cdot 2} + \frac{- 6 \cdot 16}{16}$

Step 3.2.2.6

Reorder the factors of $8 \cdot 2$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{2 \cdot 8} + \frac{- 6 \cdot 16}{16}$

Step 3.2.2.7

Multiply $2$ by $8$ .

$f (- \frac{5}{8}) = \frac{25}{16} - \frac{25 \cdot 2}{16} + \frac{- 6 \cdot 16}{16}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{5}{8}) = \frac{25 - 25 \cdot 2 - 6 \cdot 16}{16}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 25$ by $2$ .

$f (- \frac{5}{8}) = \frac{25 - 50 - 6 \cdot 16}{16}$

Step 3.2.4.2

Multiply $- 6$ by $16$ .

$f (- \frac{5}{8}) = \frac{25 - 50 - 96}{16}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $50$ from $25$ .

$f (- \frac{5}{8}) = \frac{- 25 - 96}{16}$

Step 3.2.5.2

Subtract $96$ from $- 25$ .

$f (- \frac{5}{8}) = \frac{- 121}{16}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{5}{8}) = - \frac{121}{16}$

Step 3.2.6

The final answer is $- \frac{121}{16}$ .

$- \frac{121}{16}$

Step 4

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

$(- \frac{5}{8}, - \frac{121}{16})$

Step 5