Question

question 4 (mandatory) (1 point)
a skateboard company models its profit with the function $p(x) = -2x^2 + 13x - 15$, where $x$ is the number, in thousands, that the company sells, and $p(x)$ is the profit in tens of thousands of dollars. how many skateboards must the company sell to break even? use factoring to solve.

a) at 1.5 and 5 skateboards

b) at 1500 and 5000 skateboards

c) at 150 and 500 skateboards

d) at 1500 and 50 000 skateboards

Explanation:

Step1: Set profit to zero

To break even, \( P(x) = 0 \), so we set \( -2x^{2}+13x - 15=0 \). Multiply both sides by -1: \( 2x^{2}-13x + 15 = 0 \).

Step2: Factor the quadratic

We need two numbers that multiply to \( 2\times15 = 30 \) and add to -13. The numbers are -10 and -3. Rewrite the middle term: \( 2x^{2}-10x - 3x + 15 = 0 \). Group: \( (2x^{2}-10x)+(-3x + 15)=0 \). Factor: \( 2x(x - 5)-3(x - 5)=0 \). Then \( (2x - 3)(x - 5)=0 \).

Step3: Solve for x

Set each factor to zero: \( 2x - 3 = 0 \) gives \( x=\frac{3}{2}=1.5 \); \( x - 5 = 0 \) gives \( x = 5 \). Since \( x \) is in thousands, \( 1.5\times1000 = 1500 \) and \( 5\times1000 = 5000 \).

Answer:

b) at 1500 and 5000 skateboards