Finite Math Examples

$- 10 x^{2} + 5 x + 8$

Step 1

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

$f_{max}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{5}{2 (- 10)}$

Step 2.2

Remove parentheses.

$x = - \frac{5}{2 (- 10)}$

Step 2.3

Simplify $- \frac{5}{2 (- 10)}$ .

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Step 2.3.1

Cancel the common factor of $5$ and $- 10$ .

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Step 2.3.1.1

Factor $5$ out of $5$ .

$x = - \frac{5 (1)}{2 (- 10)}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $2 (- 10)$ .

$x = - \frac{5 (1)}{5 (2 \cdot (- 2))}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{5 \cdot 1}{5 (2 \cdot (- 2))}$

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Multiply $2$ by $- 2$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 2.3.4

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

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.2

Multiply $\frac{1}{4}$ by $1$ .

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

Step 3

Evaluate $f (\frac{1}{4})$ .

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Step 3.1

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

$f (\frac{1}{4}) = - 10 {(\frac{1}{4})}^{2} + 5 (\frac{1}{4}) + 8$

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

Apply the product rule to $\frac{1}{4}$ .

$f (\frac{1}{4}) = - 10 \frac{1^{2}}{4^{2}} + 5 (\frac{1}{4}) + 8$

Step 3.2.1.2

One to any power is one.

$f (\frac{1}{4}) = - 10 \frac{1}{4^{2}} + 5 (\frac{1}{4}) + 8$

Step 3.2.1.3

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

$f (\frac{1}{4}) = - 10 (\frac{1}{16}) + 5 (\frac{1}{4}) + 8$

Step 3.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.1.4.1

Factor $2$ out of $- 10$ .

$f (\frac{1}{4}) = 2 (- 5) (\frac{1}{16}) + 5 (\frac{1}{4}) + 8$

Step 3.2.1.4.2

Factor $2$ out of $16$ .

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

Step 3.2.1.4.3

Cancel the common factor.

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

Step 3.2.1.4.4

Rewrite the expression.

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

Step 3.2.1.5

Combine $- 5$ and $\frac{1}{8}$ .

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

Step 3.2.1.6

Move the negative in front of the fraction.

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

Step 3.2.1.7

Combine $5$ and $\frac{1}{4}$ .

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

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{5}{4}$ by $\frac{2}{2}$ .

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

Step 3.2.2.2

Multiply $\frac{5}{4}$ by $\frac{2}{2}$ .

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

Step 3.2.2.3

Write $8$ as a fraction with denominator $1$ .

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

Step 3.2.2.4

Multiply $\frac{8}{1}$ by $\frac{8}{8}$ .

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

Step 3.2.2.5

Multiply $\frac{8}{1}$ by $\frac{8}{8}$ .

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

Step 3.2.2.6

Reorder the factors of $4 \cdot 2$ .

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

Step 3.2.2.7

Multiply $2$ by $4$ .

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $5$ by $2$ .

$f (\frac{1}{4}) = \frac{- 5 + 10 + 8 \cdot 8}{8}$

Step 3.2.4.2

Multiply $8$ by $8$ .

$f (\frac{1}{4}) = \frac{- 5 + 10 + 64}{8}$

Step 3.2.5

Simplify by adding numbers.

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Step 3.2.5.1

Add $- 5$ and $10$ .

$f (\frac{1}{4}) = \frac{5 + 64}{8}$

Step 3.2.5.2

Add $5$ and $64$ .

$f (\frac{1}{4}) = \frac{69}{8}$

Step 3.2.6

The final answer is $\frac{69}{8}$ .

$\frac{69}{8}$

Step 4

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

$(\frac{1}{4}, \frac{69}{8})$

Step 5