Finite Math Examples

$y = - 2 x^{2} + 10 x$

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 $b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

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

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Step 2.3.1

Cancel the common factor of $10$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $10$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Move the negative in front of the fraction.

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

Step 2.3.3

Multiply $- - \frac{5}{2}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.2

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

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

Step 3

Evaluate $f (\frac{5}{2})$ .

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Step 3.1

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

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

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 $\frac{5}{2}$ .

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

Step 3.2.1.2

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

$f (\frac{5}{2}) = - 2 \frac{25}{2^{2}} + 10 (\frac{5}{2})$

Step 3.2.1.3

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

$f (\frac{5}{2}) = - 2 (\frac{25}{4}) + 10 (\frac{5}{2})$

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 $- 2$ .

$f (\frac{5}{2}) = 2 (- 1) (\frac{25}{4}) + 10 (\frac{5}{2})$

Step 3.2.1.4.2

Factor $2$ out of $4$ .

$f (\frac{5}{2}) = 2 \cdot (- 1 \frac{25}{2 \cdot 2}) + 10 (\frac{5}{2})$

Step 3.2.1.4.3

Cancel the common factor.

$f (\frac{5}{2}) = 2 \cdot (- 1 \frac{25}{2 \cdot 2}) + 10 (\frac{5}{2})$

Step 3.2.1.4.4

Rewrite the expression.

$f (\frac{5}{2}) = - 1 (\frac{25}{2}) + 10 (\frac{5}{2})$

Step 3.2.1.5

Rewrite $- 1 (\frac{25}{2})$ as $- (\frac{25}{2})$ .

$f (\frac{5}{2}) = - (\frac{25}{2}) + 10 (\frac{5}{2})$

Step 3.2.1.6

Cancel the common factor of $2$ .

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Step 3.2.1.6.1

Factor $2$ out of $10$ .

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

Step 3.2.1.6.2

Cancel the common factor.

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

Step 3.2.1.6.3

Rewrite the expression.

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

Step 3.2.1.7

Multiply $5$ by $5$ .

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

Step 3.2.2

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

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

Step 3.2.3

Combine $25$ and $\frac{2}{2}$ .

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $25$ by $2$ .

$f (\frac{5}{2}) = \frac{- 25 + 50}{2}$

Step 3.2.5.2

Add $- 25$ and $50$ .

$f (\frac{5}{2}) = \frac{25}{2}$

Step 3.2.6

The final answer is $\frac{25}{2}$ .

$\frac{25}{2}$

Step 4

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

$(\frac{5}{2}, \frac{25}{2})$

Step 5