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An online retailer models its weekly profit P, in thousands of dollars, from selling x hundred phone cases as P = 12x − x² − 32. At the lower of the two sales levels where weekly profit equals zero, how many hundred phone cases does the retailer sell?
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Correct answer
B
Method 1 (factoring). Set profit to zero: 12x − x² − 32 = 0. Multiply through by −1 to get the standard form x² − 12x + 32 = 0. Look for two numbers that multiply to 32 and add to 12: those are 4 and 8, so the equation factors as (x − 4)(x − 8) = 0, giving roots x = 4 and x = 8. As sales rise from zero, profit starts negative (at x = 0 it is −32), first reaches zero at the lower root x = 4, then peaks, then returns to zero at x = 8. The lower break-even level is x = 4, so the answer is (B).
Method 2 (quadratic formula). For x² − 12x + 32 = 0, the discriminant is 12² − 4 × 32 = 144 − 128 = 16, and √16 = 4. The roots are (12 ± 4) ÷ 2, which gives 8 and 4. The smaller root is 4, confirming (B).
Why the wrong choices tempt: (A) 2 comes from grabbing the factor pair 2 and 16 of 32 while ignoring that the factors must sum to 12 (2 + 16 is 18, not 12). (C) 6 is the axis of symmetry, x = 12 ÷ 2, which locates the profit peak, not a break-even point. (D) 8 is the correct partner root but it is the higher break-even level, while the stem asks for the lower one. (E) 16 results from a sign slip when removing the leading negative: dropping the sign on the constant produces x² − 12x − 32 = 0, whose positive root is 16. Only x = 4 satisfies the original equation and the request for the lower break-even level.
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