Find the Maximum/Minimum Value -3x^2-18x-24

$- 3 x^{2} - 18 x - 24$

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 $b$ .

$x = - \frac{- 18}{2 (- 3)}$

Remove parentheses.

$x = - \frac{- 18}{2 (- 3)}$

Simplify $- \frac{- 18}{2 (- 3)}$ .

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

$x = - \frac{2 \cdot - 9}{2 \cdot - 3}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 9}{2 (- 3)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 9}{2 \cdot - 3}$

Rewrite the expression.

$x = - \frac{- 9}{- 3}$

Cancel the common factor of $- 9$ and $- 3$ .

Factor $- 3$ out of $- 9$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

$x = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x = - 3$

Step 3

Evaluate $f (- 3)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 3 {(- 3)}^{2} - 18 \cdot - 3 - 24$

Simplify the result.

Simplify each term.

Multiply $- 3$ by ${(- 3)}^{2}$ by adding the exponents.

Multiply $- 3$ by ${(- 3)}^{2}$ .

Raise $- 3$ to the power of $1$ .

$f (- 3) = (- 3) {(- 3)}^{2} - 18 \cdot - 3 - 24$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 3) = {(- 3)}^{1 + 2} - 18 \cdot - 3 - 24$

Add $1$ and $2$ .

$f (- 3) = {(- 3)}^{3} - 18 \cdot - 3 - 24$

Raise $- 3$ to the power of $3$ .

$f (- 3) = - 27 - 18 \cdot - 3 - 24$

Multiply $- 18$ by $- 3$ .

$f (- 3) = - 27 + 54 - 24$

Simplify by adding and subtracting.

Add $- 27$ and $54$ .

$f (- 3) = 27 - 24$

Subtract $24$ from $27$ .

$f (- 3) = 3$

The final answer is $3$ .

$3$

Step 4

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

$(- 3, 3)$

Step 5

Find the Maximum/Minimum Value -3x^2-18x-24

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