PSAT Math Multiple-Choice Question 677: Answer and Explanation

Question: 677

The supply for a given item at a varying price p (in dollars) is given by the equation s(p) = 3p + 6p². The demand for the same item at a varying price p is given by the equation d(p) = 156 - 12p. At what price are the supply and the demand for the item equivalent?

• A. $3.50
• B. $4
• C. $6.50
• D. $12

Correct Answer: B

Explanation:

(B) Find where the supply and demand are equivalent by setting the two equations equal to one another and solving for p:

3p + 6p² = 156 - 12p

Subtract 156 and add 12p to both sides:

15p + 6p² - 156 = 0

Rearrange the equation to get it in ax² + bx + c form while simultaneously factoring out a 3:

3(2p² + 5p - 52) = 0

Divide both sides by 3:

2p² + 5p - 52 = 0

Factor to get:

(2p + 13)(p - 4) = 0

Set each factor equal to 0 and solve for p to get the two possible values of p:

p - 4 = 0 so p = 4

In this situation, p must be positive since it represents the price of the item, which can't be negative. Therefore, p equals only 4, choice (B).

Alternatively, if you didn't recognize that the quadratic equation could be factored, you could have used the quadratic formula: