Find the Maximum/Minimum Value y=-4x^2-24x+-36

$y = - 4 x^{2} - 24 x + - 36$

Step 1

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

$y = - 4 x^{2} - 24 x - 36$

Step 2

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 3

Find the value of $x = - \frac{b}{2 a}$ .

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

$x = - \frac{- 24}{2 (- 4)}$

Remove parentheses.

$x = - \frac{- 24}{2 (- 4)}$

Simplify $- \frac{- 24}{2 (- 4)}$ .

Cancel the common factor of $- 24$ and $2$ .

Factor $2$ out of $- 24$ .

$x = - \frac{2 \cdot - 12}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$x = - \frac{2 \cdot - 12}{2 (- 4)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 12}{2 \cdot - 4}$

Rewrite the expression.

$x = - \frac{- 12}{- 4}$

Cancel the common factor of $- 12$ and $- 4$ .

Factor $- 4$ out of $- 12$ .

$x = - \frac{- 4 (3)}{- 4}$

Cancel the common factors.

Factor $- 4$ out of $- 4$ .

$x = - \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Cancel the common factor.

$x = - \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Rewrite the expression.

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

Divide $3$ by $1$ .

$x = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x = - 3$

Step 4

Evaluate $f (- 3)$ .

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

$f (- 3) = - 4 {(- 3)}^{2} - 24 \cdot - 3 - 36$

Simplify the result.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$f (- 3) = - 4 \cdot 9 - 24 \cdot - 3 - 36$

Multiply $- 4$ by $9$ .

$f (- 3) = - 36 - 24 \cdot - 3 - 36$

Multiply $- 24$ by $- 3$ .

$f (- 3) = - 36 + 72 - 36$

Simplify by adding and subtracting.

Add $- 36$ and $72$ .

$f (- 3) = 36 - 36$

Subtract $36$ from $36$ .

$f (- 3) = 0$

The final answer is $0$ .

$0$

Step 5

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

$(- 3, 0)$

Step 6

Find the Maximum/Minimum Value y=-4x^2-24x+-36

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