Algebra Examples

$h (t) = 52 t - 16 t^{2}$

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 $b$ .

$t = - \frac{52}{2 (- 16)}$

Step 2.2

Remove parentheses.

$t = - \frac{52}{2 (- 16)}$

Step 2.3

Simplify $- \frac{52}{2 (- 16)}$ .

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Step 2.3.1

Cancel the common factor of $52$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $52$ .

$t = - \frac{2 \cdot 26}{2 \cdot - 16}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

$t = - \frac{2 \cdot 26}{2 (- 16)}$

Step 2.3.1.2.2

Cancel the common factor.

$t = - \frac{2 \cdot 26}{2 \cdot - 16}$

Step 2.3.1.2.3

Rewrite the expression.

$t = - \frac{26}{- 16}$

Step 2.3.2

Cancel the common factor of $26$ and $- 16$ .

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Step 2.3.2.1

Factor $2$ out of $26$ .

$t = - \frac{2 (13)}{- 16}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $2$ out of $- 16$ .

$t = - \frac{2 \cdot 13}{2 \cdot - 8}$

Step 2.3.2.2.2

Cancel the common factor.

$t = - \frac{2 \cdot 13}{2 \cdot - 8}$

Step 2.3.2.2.3

Rewrite the expression.

$t = - \frac{13}{- 8}$

Step 2.3.3

Move the negative in front of the fraction.

$t = - - \frac{13}{8}$

Step 2.3.4

Multiply $- - \frac{13}{8}$ .

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{13}{8})$

Step 2.3.4.2

Multiply $\frac{13}{8}$ by $1$ .

$t = \frac{13}{8}$

Step 3

Evaluate $f (\frac{13}{8})$ .

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Step 3.1

Replace the variable $t$ with $\frac{13}{8}$ in the expression.

$h (\frac{13}{8}) = 52 (\frac{13}{8}) - 16 {(\frac{13}{8})}^{2}$

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

Cancel the common factor of $4$ .

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Step 3.2.1.1.1

Factor $4$ out of $52$ .

$h (\frac{13}{8}) = 4 (13) (\frac{13}{8}) - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.1.2

Factor $4$ out of $8$ .

$h (\frac{13}{8}) = 4 \cdot (13 (\frac{13}{4 \cdot 2})) - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.1.3

Cancel the common factor.

$h (\frac{13}{8}) = 4 \cdot (13 (\frac{13}{4 \cdot 2})) - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.1.4

Rewrite the expression.

$h (\frac{13}{8}) = 13 (\frac{13}{2}) - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.2

Combine $13$ and $\frac{13}{2}$ .

$h (\frac{13}{8}) = \frac{13 \cdot 13}{2} - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.3

Multiply $13$ by $13$ .

$h (\frac{13}{8}) = \frac{169}{2} - 16 {(\frac{13}{8})}^{2}$

Step 3.2.1.4

Apply the product rule to $\frac{13}{8}$ .

$h (\frac{13}{8}) = \frac{169}{2} - 16 \frac{13^{2}}{8^{2}}$

Step 3.2.1.5

Raise $13$ to the power of $2$ .

$h (\frac{13}{8}) = \frac{169}{2} - 16 \frac{169}{8^{2}}$

Step 3.2.1.6

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

$h (\frac{13}{8}) = \frac{169}{2} - 16 (\frac{169}{64})$

Step 3.2.1.7

Cancel the common factor of $16$ .

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Step 3.2.1.7.1

Factor $16$ out of $- 16$ .

$h (\frac{13}{8}) = \frac{169}{2} + 16 (- 1) (\frac{169}{64})$

Step 3.2.1.7.2

Factor $16$ out of $64$ .

$h (\frac{13}{8}) = \frac{169}{2} + 16 \cdot (- 1 \frac{169}{16 \cdot 4})$

Step 3.2.1.7.3

Cancel the common factor.

$h (\frac{13}{8}) = \frac{169}{2} + 16 \cdot (- 1 \frac{169}{16 \cdot 4})$

Step 3.2.1.7.4

Rewrite the expression.

$h (\frac{13}{8}) = \frac{169}{2} - 1 (\frac{169}{4})$

Step 3.2.1.8

Rewrite $- 1 (\frac{169}{4})$ as $- (\frac{169}{4})$ .

$h (\frac{13}{8}) = \frac{169}{2} - \frac{169}{4}$

Step 3.2.2

To write $\frac{169}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$h (\frac{13}{8}) = \frac{169}{2} \cdot \frac{2}{2} - \frac{169}{4}$

Step 3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{169}{2}$ by $\frac{2}{2}$ .

$h (\frac{13}{8}) = \frac{169 \cdot 2}{2 \cdot 2} - \frac{169}{4}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$h (\frac{13}{8}) = \frac{169 \cdot 2}{4} - \frac{169}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$h (\frac{13}{8}) = \frac{169 \cdot 2 - 169}{4}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $169$ by $2$ .

$h (\frac{13}{8}) = \frac{338 - 169}{4}$

Step 3.2.5.2

Subtract $169$ from $338$ .

$h (\frac{13}{8}) = \frac{169}{4}$

Step 3.2.6

The final answer is $\frac{169}{4}$ .

$\frac{169}{4}$

Step 4

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

$(\frac{13}{8}, \frac{169}{4})$

Step 5