Algebra Examples

$d (t) = - 16 t^{2} + 32 t + 40$

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 $b$ .

$t = - \frac{32}{2 (- 16)}$

Step 2.2

Remove parentheses.

$t = - \frac{32}{2 (- 16)}$

Step 2.3

Simplify $- \frac{32}{2 (- 16)}$ .

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Step 2.3.1

Cancel the common factor of $32$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $32$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $16$ out of $16$ .

$t = - \frac{16 (1)}{- 16}$

Step 2.3.2.2

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$t = - (- 1 \cdot 1)$

Step 2.3.3

Multiply $- (- 1 \cdot 1)$ .

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Step 2.3.3.1

Multiply $- 1$ by $1$ .

$t = - - 1$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$t = 1$

Step 3

Evaluate $f (1)$ .

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Step 3.1

Replace the variable $t$ with $1$ in the expression.

$d (1) = - 16 {(1)}^{2} + 32 (1) + 40$

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

One to any power is one.

$d (1) = - 16 \cdot 1 + 32 (1) + 40$

Step 3.2.1.2

Multiply $- 16$ by $1$ .

$d (1) = - 16 + 32 (1) + 40$

Step 3.2.1.3

Multiply $32$ by $1$ .

$d (1) = - 16 + 32 + 40$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 16$ and $32$ .

$d (1) = 16 + 40$

Step 3.2.2.2

Add $16$ and $40$ .

$d (1) = 56$

Step 3.2.3

The final answer is $56$ .

$56$

Step 4

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

$(1, 56)$

Step 5