Find the Maximum/Minimum Value w(x)=-5(x-8)(x+4)

$w (x) = - 5 (x - 8) (x + 4)$

Step 1

Simplify by multiplying through.

Apply the distributive property.

$w (x) = (- 5 x - 5 \cdot - 8) (x + 4)$

Multiply $- 5$ by $- 8$ .

$w (x) = (- 5 x + 40) (x + 4)$

Step 2

Expand $(- 5 x + 40) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$w (x) = - 5 x (x + 4) + 40 (x + 4)$

Apply the distributive property.

$w (x) = - 5 x \cdot x - 5 x \cdot 4 + 40 (x + 4)$

Apply the distributive property.

$w (x) = - 5 x \cdot x - 5 x \cdot 4 + 40 x + 40 \cdot 4$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$w (x) = - 5 (x \cdot x) - 5 x \cdot 4 + 40 x + 40 \cdot 4$

Multiply $x$ by $x$ .

$w (x) = - 5 x^{2} - 5 x \cdot 4 + 40 x + 40 \cdot 4$

Multiply $4$ by $- 5$ .

$w (x) = - 5 x^{2} - 20 x + 40 x + 40 \cdot 4$

Multiply $40$ by $4$ .

$w (x) = - 5 x^{2} - 20 x + 40 x + 160$

Add $- 20 x$ and $40 x$ .

$w (x) = - 5 x^{2} + 20 x + 160$

Step 4

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 5

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

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

$x = - \frac{20}{2 (- 5)}$

Remove parentheses.

$x = - \frac{20}{2 (- 5)}$

Simplify $- \frac{20}{2 (- 5)}$ .

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$x = - \frac{2 \cdot 10}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$x = - \frac{2 \cdot 10}{2 (- 5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 10}{2 \cdot - 5}$

Rewrite the expression.

$x = - \frac{10}{- 5}$

Cancel the common factor of $10$ and $- 5$ .

Factor $5$ out of $10$ .

$x = - \frac{5 (2)}{- 5}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$x = - - 2$

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 6

Evaluate $f (2)$ .

Replace the variable $x$ with $2$ in the expression.

$w (2) = - 5 {(2)}^{2} + 20 (2) + 160$

Simplify the result.

Simplify each term.

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

$w (2) = - 5 \cdot 4 + 20 (2) + 160$

Multiply $- 5$ by $4$ .

$w (2) = - 20 + 20 (2) + 160$

Multiply $20$ by $2$ .

$w (2) = - 20 + 40 + 160$

Simplify by adding numbers.

Add $- 20$ and $40$ .

$w (2) = 20 + 160$

Add $20$ and $160$ .

$w (2) = 180$

The final answer is $180$ .

$180$

Step 7

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

$(2, 180)$

Step 8

Find the Maximum/Minimum Value w(x)=-5(x-8)(x+4)

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