Question

a companys profit p (in thousands of dollars) is modeled by $p = -2x^2 + 20x - 18$, where $x$ is the number of items sold (in hundreds). when is the profit equal to 32?
a) 400 items
b) 450 items
c) 475 items
d) 500 items
set up your equation and convert to standard form
$32=-2x^2 + 20x - 18$
$-32$ $-32$
$-2x^2 + 20x$
now use gcf and factoring to solve the story problem:

Explanation:

Step1: Set up the equation

We know that \( P = 32 \), so substitute \( P \) into the profit function:
\( 32=-2x^{2}+20x - 18 \)
Then, move all terms to one side to get the standard form of a quadratic equation (\( ax^{2}+bx + c = 0 \)):
\( 2x^{2}-20x + 50 = 0 \) (We multiply both sides by - 1 to make the coefficient of \( x^{2} \) positive: \( - 32=2x^{2}-20x + 18 \), then \( 2x^{2}-20x+18 + 32=0 \), so \( 2x^{2}-20x + 50 = 0 \))

Step2: Find the GCF

The greatest common factor (GCF) of the coefficients \( 2\), \( - 20 \) and \( 50 \) is \( 2 \). Divide each term by \( 2 \):
\( \frac{2x^{2}}{2}-\frac{20x}{2}+\frac{50}{2}=0\div2 \)
\( x^{2}-10x + 25 = 0 \)

Step3: Factor the quadratic

The quadratic \( x^{2}-10x + 25 \) is a perfect square trinomial, which factors as \( (x - 5)^{2}=0 \) (since \( (a - b)^{2}=a^{2}-2ab + b^{2} \), here \( a=x \), \( b = 5 \), and \( - 2ab=-10x \), \( b^{2}=25 \))

Step4: Solve for x

Set \( (x - 5)^{2}=0 \), then \( x-5 = 0 \), so \( x = 5 \)

Step5: Interpret the value of x

Since \( x \) is the number of items sold in hundreds, the number of items is \( x\times100 \). Substitute \( x = 5 \):
Number of items \(=5\times100 = 500\)

Answer:

D. 500 items