value in every choice is 1. So and . If , then b = –8. If
, then and 4 i^2 = 64 – 4c.Finally, 4 c = 64 – 4i^2 and i^2 = –1, so 4 c = 64 + 4, 4c = 68, and c = 17. Collecting these values of a,
b, and c, ax^2 + bx + c = 0 is x^2 – 8x + 17 = 0.Another way to approach this problem is to generate the expression from the factors. If 4 + i
and 4 – i are roots, then [x − (4 + i)] and [x − (4 − i)] are factors. Multiplying these together will
result in the full expression:Notice all the terms with just i cancel out, leaving x^2 − 8x +16 − i^2 . But since i^2 = −1, this
becomes x^2 − 8x +16 − (−1) = x^2 − 8x +17.3 . A
The statement “all Martians vacation on Venus,” is logically equivalent to the statement “If
Martian, then vacation on Venus.” The contrapositive of this statement is “If no vacation on
Venus, then not a Martian”—which is just about identical to (A).
4 . B
Translate carefully. If “Anselm is three times as old as Bartholomew,” then A = 3B. And if
“Bartholomew is 4 years younger than Catherine,” then B = C – 4. Consider the statements
one by one, Picking Numbers as you go. Must it be true, as I has it, that in five years, A = 3 B?
No. Say A = 3 and B = 1. Then in five years, A = 8 and B = 6, and 8 ≠ 3(6). Eliminate (A) and (D).
Must it be true, as II has it, that in five years, C = B + 4? It must, as any numbers you pick will
demonstrate. Siblings who are four years apart now will always be four years apart.
Eliminate (C). Must it be true, as III has it, that A > C? No, it could be true—if, for example A =
90 , then B = 30 and C = 34. But it doesn’t have to be true—as is shown, for example, when A =
3 ; then B = 1 and C = 5. In five years, A = 8 and C = 10. Eliminate (E).
5 . D