Finite Math Examples

$- 16 (x^{2} - 9 x)$

Step 1

Apply the distributive property.

$f (x) = - 16 x^{2} - 16 (- 9 x)$

Step 2

Multiply $- 9$ by $- 16$ .

$f (x) = - 16 x^{2} + 144 x$

Step 3

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 4

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

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Step 4.1

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

$x = - \frac{144}{2 (- 16)}$

Step 4.2

Remove parentheses.

$x = - \frac{144}{2 (- 16)}$

Step 4.3

Simplify $- \frac{144}{2 (- 16)}$ .

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Step 4.3.1

Cancel the common factor of $144$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $144$ .

$x = - \frac{2 \cdot 72}{2 \cdot - 16}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 \cdot - 16$ .

$x = - \frac{2 \cdot 72}{2 (- 16)}$

Step 4.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 72}{2 \cdot - 16}$

Step 4.3.1.2.3

Rewrite the expression.

$x = - \frac{72}{- 16}$

Step 4.3.2

Cancel the common factor of $72$ and $- 16$ .

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Step 4.3.2.1

Factor $8$ out of $72$ .

$x = - \frac{8 (9)}{- 16}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $8$ out of $- 16$ .

$x = - \frac{8 \cdot 9}{8 \cdot - 2}$

Step 4.3.2.2.2

Cancel the common factor.

$x = - \frac{8 \cdot 9}{8 \cdot - 2}$

Step 4.3.2.2.3

Rewrite the expression.

$x = - \frac{9}{- 2}$

Step 4.3.3

Move the negative in front of the fraction.

$x = - - \frac{9}{2}$

Step 4.3.4

Multiply $- - \frac{9}{2}$ .

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Step 4.3.4.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{9}{2})$

Step 4.3.4.2

Multiply $\frac{9}{2}$ by $1$ .

$x = \frac{9}{2}$

Step 5

Evaluate $f (\frac{9}{2})$ .

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Step 5.1

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

$f (\frac{9}{2}) = - 16 {(\frac{9}{2})}^{2} + 144 (\frac{9}{2})$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{9}{2}$ .

$f (\frac{9}{2}) = - 16 \frac{9^{2}}{2^{2}} + 144 (\frac{9}{2})$

Step 5.2.1.2

Raise $9$ to the power of $2$ .

$f (\frac{9}{2}) = - 16 \frac{81}{2^{2}} + 144 (\frac{9}{2})$

Step 5.2.1.3

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

$f (\frac{9}{2}) = - 16 (\frac{81}{4}) + 144 (\frac{9}{2})$

Step 5.2.1.4

Cancel the common factor of $4$ .

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Step 5.2.1.4.1

Factor $4$ out of $- 16$ .

$f (\frac{9}{2}) = 4 (- 4) (\frac{81}{4}) + 144 (\frac{9}{2})$

Step 5.2.1.4.2

Cancel the common factor.

$f (\frac{9}{2}) = 4 \cdot (- 4 (\frac{81}{4})) + 144 (\frac{9}{2})$

Step 5.2.1.4.3

Rewrite the expression.

$f (\frac{9}{2}) = - 4 \cdot 81 + 144 (\frac{9}{2})$

Step 5.2.1.5

Multiply $- 4$ by $81$ .

$f (\frac{9}{2}) = - 324 + 144 (\frac{9}{2})$

Step 5.2.1.6

Cancel the common factor of $2$ .

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Step 5.2.1.6.1

Factor $2$ out of $144$ .

$f (\frac{9}{2}) = - 324 + 2 (72) (\frac{9}{2})$

Step 5.2.1.6.2

Cancel the common factor.

$f (\frac{9}{2}) = - 324 + 2 \cdot (72 (\frac{9}{2}))$

Step 5.2.1.6.3

Rewrite the expression.

$f (\frac{9}{2}) = - 324 + 72 \cdot 9$

Step 5.2.1.7

Multiply $72$ by $9$ .

$f (\frac{9}{2}) = - 324 + 648$

Step 5.2.2

Add $- 324$ and $648$ .

$f (\frac{9}{2}) = 324$

Step 5.2.3

The final answer is $324$ .

$324$

Step 6

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

$(\frac{9}{2}, 324)$

Step 7