Finite Math Examples

$f (x) = - 3 x^{2} - 30 x - 74$

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 $b$ .

$x = - \frac{- 30}{2 (- 3)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 30}{2 (- 3)}$

Step 2.3

Simplify $- \frac{- 30}{2 (- 3)}$ .

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Step 2.3.1

Cancel the common factor of $- 30$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 30$ .

$x = - \frac{2 \cdot - 15}{2 \cdot - 3}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 3$ .

$x = - \frac{2 \cdot - 15}{2 (- 3)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 15}{2 \cdot - 3}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 15}{- 3}$

Step 2.3.2

Cancel the common factor of $- 15$ and $- 3$ .

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Step 2.3.2.1

Factor $- 3$ out of $- 15$ .

$x = - \frac{- 3 (5)}{- 3}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $- 3$ out of $- 3$ .

$x = - \frac{- 3 \cdot 5}{- 3 \cdot 1}$

Step 2.3.2.2.2

Cancel the common factor.

$x = - \frac{- 3 \cdot 5}{- 3 \cdot 1}$

Step 2.3.2.2.3

Rewrite the expression.

$x = - \frac{5}{1}$

Step 2.3.2.2.4

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Step 2.3.3

Multiply $- 1$ by $5$ .

$x = - 5$

Step 3

Evaluate $f (- 5)$ .

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Step 3.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - 3 {(- 5)}^{2} - 30 \cdot - 5 - 74$

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

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

$f (- 5) = - 3 \cdot 25 - 30 \cdot - 5 - 74$

Step 3.2.1.2

Multiply $- 3$ by $25$ .

$f (- 5) = - 75 - 30 \cdot - 5 - 74$

Step 3.2.1.3

Multiply $- 30$ by $- 5$ .

$f (- 5) = - 75 + 150 - 74$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 75$ and $150$ .

$f (- 5) = 75 - 74$

Step 3.2.2.2

Subtract $74$ from $75$ .

$f (- 5) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

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

$(- 5, 1)$

Step 5