Algebra Examples

$- 16 t^{2} + 85 t$

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 $f (- \frac{b}{2 a})$ .

$f_{max}$ $t = a t^{2} + b t + c$ occurs at $t = - \frac{b}{2 a}$

Step 2

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

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Step 2.1

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

$t = - \frac{85}{2 (- 16)}$

Step 2.2

Remove parentheses.

$t = - \frac{85}{2 (- 16)}$

Step 2.3

Simplify $- \frac{85}{2 (- 16)}$ .

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Step 2.3.1

Multiply $2$ by $- 16$ .

$t = - \frac{85}{- 32}$

Step 2.3.2

Move the negative in front of the fraction.

$t = - - \frac{85}{32}$

Step 2.3.3

Multiply $- - \frac{85}{32}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{85}{32})$

Step 2.3.3.2

Multiply $\frac{85}{32}$ by $1$ .

$t = \frac{85}{32}$

Step 3

Evaluate $f (\frac{85}{32})$ .

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Step 3.1

Replace the variable $t$ with $\frac{85}{32}$ in the expression.

$f (\frac{85}{32}) = - 16 {(\frac{85}{32})}^{2} + 85 (\frac{85}{32})$

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 $\frac{85}{32}$ .

$f (\frac{85}{32}) = - 16 \frac{85^{2}}{32^{2}} + 85 (\frac{85}{32})$

Step 3.2.1.2

Raise $85$ to the power of $2$ .

$f (\frac{85}{32}) = - 16 \frac{7225}{32^{2}} + 85 (\frac{85}{32})$

Step 3.2.1.3

Raise $32$ to the power of $2$ .

$f (\frac{85}{32}) = - 16 (\frac{7225}{1024}) + 85 (\frac{85}{32})$

Step 3.2.1.4

Cancel the common factor of $16$ .

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Step 3.2.1.4.1

Factor $16$ out of $- 16$ .

$f (\frac{85}{32}) = 16 (- 1) (\frac{7225}{1024}) + 85 (\frac{85}{32})$

Step 3.2.1.4.2

Factor $16$ out of $1024$ .

$f (\frac{85}{32}) = 16 \cdot (- 1 \frac{7225}{16 \cdot 64}) + 85 (\frac{85}{32})$

Step 3.2.1.4.3

Cancel the common factor.

$f (\frac{85}{32}) = 16 \cdot (- 1 \frac{7225}{16 \cdot 64}) + 85 (\frac{85}{32})$

Step 3.2.1.4.4

Rewrite the expression.

$f (\frac{85}{32}) = - 1 (\frac{7225}{64}) + 85 (\frac{85}{32})$

Step 3.2.1.5

Rewrite $- 1 (\frac{7225}{64})$ as $- (\frac{7225}{64})$ .

$f (\frac{85}{32}) = - (\frac{7225}{64}) + 85 (\frac{85}{32})$

Step 3.2.1.6

Multiply $85 (\frac{85}{32})$ .

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Step 3.2.1.6.1

Combine $85$ and $\frac{85}{32}$ .

$f (\frac{85}{32}) = - \frac{7225}{64} + \frac{85 \cdot 85}{32}$

Step 3.2.1.6.2

Multiply $85$ by $85$ .

$f (\frac{85}{32}) = - \frac{7225}{64} + \frac{7225}{32}$

Step 3.2.2

To write $\frac{7225}{32}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{85}{32}) = - \frac{7225}{64} + \frac{7225}{32} \cdot \frac{2}{2}$

Step 3.2.3

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{7225}{32}$ by $\frac{2}{2}$ .

$f (\frac{85}{32}) = - \frac{7225}{64} + \frac{7225 \cdot 2}{32 \cdot 2}$

Step 3.2.3.2

Multiply $32$ by $2$ .

$f (\frac{85}{32}) = - \frac{7225}{64} + \frac{7225 \cdot 2}{64}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{85}{32}) = \frac{- 7225 + 7225 \cdot 2}{64}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $7225$ by $2$ .

$f (\frac{85}{32}) = \frac{- 7225 + 14450}{64}$

Step 3.2.5.2

Add $- 7225$ and $14450$ .

$f (\frac{85}{32}) = \frac{7225}{64}$

Step 3.2.6

The final answer is $\frac{7225}{64}$ .

$\frac{7225}{64}$

Step 4

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

$(\frac{85}{32}, \frac{7225}{64})$

Step 5