Find the Maximum/Minimum Value p(x)=-2(x-9)^2+200

$p (x) = - 2 {(x - 9)}^{2} + 200$

Step 1

Simplify each term.

Rewrite ${(x - 9)}^{2}$ as $(x - 9) (x - 9)$ .

$p (x) = - 2 ((x - 9) (x - 9)) + 200$

Expand $(x - 9) (x - 9)$ using the FOIL Method.

Apply the distributive property.

$p (x) = - 2 (x (x - 9) - 9 (x - 9)) + 200$

Apply the distributive property.

$p (x) = - 2 (x \cdot x + x \cdot - 9 - 9 (x - 9)) + 200$

Apply the distributive property.

$p (x) = - 2 (x \cdot x + x \cdot - 9 - 9 x - 9 \cdot - 9) + 200$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$p (x) = - 2 (x^{2} + x \cdot - 9 - 9 x - 9 \cdot - 9) + 200$

Move $- 9$ to the left of $x$ .

$p (x) = - 2 (x^{2} - 9 \cdot x - 9 x - 9 \cdot - 9) + 200$

Multiply $- 9$ by $- 9$ .

$p (x) = - 2 (x^{2} - 9 x - 9 x + 81) + 200$

Subtract $9 x$ from $- 9 x$ .

$p (x) = - 2 (x^{2} - 18 x + 81) + 200$

Apply the distributive property.

$p (x) = - 2 x^{2} - 2 (- 18 x) - 2 \cdot 81 + 200$

Simplify.

Multiply $- 18$ by $- 2$ .

$p (x) = - 2 x^{2} + 36 x - 2 \cdot 81 + 200$

Multiply $- 2$ by $81$ .

$p (x) = - 2 x^{2} + 36 x - 162 + 200$

Step 2

Add $- 162$ and $200$ .

$p (x) = - 2 x^{2} + 36 x + 38$

Step 3

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 4

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

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

$x = - \frac{36}{2 (- 2)}$

Remove parentheses.

$x = - \frac{36}{2 (- 2)}$

Simplify $- \frac{36}{2 (- 2)}$ .

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$x = - \frac{2 \cdot 18}{2 \cdot - 2}$

Cancel the common factors.

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

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

Cancel the common factor.

$x = - \frac{2 \cdot 18}{2 \cdot - 2}$

Rewrite the expression.

$x = - \frac{18}{- 2}$

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

Factor $2$ out of $18$ .

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

Move the negative one from the denominator of $\frac{9}{- 1}$ .

$x = - (- 1 \cdot 9)$

Multiply $- (- 1 \cdot 9)$ .

Multiply $- 1$ by $9$ .

$x = - - 9$

Multiply $- 1$ by $- 9$ .

$x = 9$

Step 5

Evaluate $f (9)$ .

Replace the variable $x$ with $9$ in the expression.

$p (9) = - 2 {(9)}^{2} + 36 (9) + 38$

Simplify the result.

Simplify each term.

Raise $9$ to the power of $2$ .

$p (9) = - 2 \cdot 81 + 36 (9) + 38$

Multiply $- 2$ by $81$ .

$p (9) = - 162 + 36 (9) + 38$

Multiply $36$ by $9$ .

$p (9) = - 162 + 324 + 38$

Simplify by adding numbers.

Add $- 162$ and $324$ .

$p (9) = 162 + 38$

Add $162$ and $38$ .

$p (9) = 200$

The final answer is $200$ .

$200$

Step 6

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

$(9, 200)$

Step 7

Find the Maximum/Minimum Value p(x)=-2(x-9)^2+200

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