Find the Maximum/Minimum Value g(x)=-3x^2-30x-72

$g (x) = - 3 x^{2} - 30 x - 72$

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 $x = - \frac{b}{2 a}$ .

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

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

Remove parentheses.

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

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

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

Factor $2$ out of $- 30$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$x = - 5$

Step 3

Evaluate $f (- 5)$ .

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

$g (- 5) = - 3 {(- 5)}^{2} - 30 \cdot - 5 - 72$

Simplify the result.

Simplify each term.

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

$g (- 5) = - 3 \cdot 25 - 30 \cdot - 5 - 72$

Multiply $- 3$ by $25$ .

$g (- 5) = - 75 - 30 \cdot - 5 - 72$

Multiply $- 30$ by $- 5$ .

$g (- 5) = - 75 + 150 - 72$

Simplify by adding and subtracting.

Add $- 75$ and $150$ .

$g (- 5) = 75 - 72$

Subtract $72$ from $75$ .

$g (- 5) = 3$

The final answer is $3$ .

$3$

Step 4

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

$(- 5, 3)$

Step 5

Find the Maximum/Minimum Value g(x)=-3x^2-30x-72

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