Does x = 9 - bx?
(1) x = b
(2) b = -1
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prachi18oct
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If x=b, then our question becomes:prachi18oct wrote:Does x = 9 - bx?
(1) x = b
(2) b = -1
"Does x = 9 - x^2 ? "
or rewriting, the question becomes
"Does x^2 + x - 9 = 0? "
The answer to that question is almost always 'no' (for example, if x=0, the answer is 'no'). The answer can also be 'yes', since that quadratic does indeed have two solutions. One way to see that is to use the discriminant from the quadratic formula (something you'll never need on a real GMAT question) - the equation ax^2 + bx + c = 0 will have two distinct solutions if b^2 - 4ac > 0, which is true here. There are other ways to see that this equation has solutions, but all use math you never need on the GMAT (properties of parabolas in coordinate geometry, or the intermediate value theorem from calculus). It's not the kind of equation you'd see on the GMAT, and it would be more realistic if the quadratic had a simple factorization, so we could see using simple algebra that the answer is sometimes yes, sometimes no. Regardless, it's not sufficient.
When we use Statement 2, we can replace b with -1 to rephrase our question:
"Does x = 9 - (-1)x ?"
"Does x = 9 + x ?"
"Does 0 = 9 ?"
and clearly the answer to that question is 'no', so Statement 2 is sufficient. The answer is B.
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theCEO
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Thanks for the detailed explanation Ian. Post editedIan Stewart wrote:If x=b, then our question becomes:prachi18oct wrote:Does x = 9 - bx?
(1) x = b
(2) b = -1
"Does x = 9 - x^2 ? "
or rewriting, the question becomes
"Does x^2 + x - 9 = 0? "
The answer to that question is almost always 'no' (for example, if x=0, the answer is 'no'). The answer can also be 'yes', since that quadratic does indeed have two solutions. One way to see that is to use the discriminant from the quadratic formula (something you'll never need on a real GMAT question) - the equation ax^2 + bx + c = 0 will have two distinct solutions if b^2 - 4ac > 0, which is true here. There are other ways to see that this equation has solutions, but all use math you never need on the GMAT (properties of parabolas in coordinate geometry, or the intermediate value theorem from calculus). It's not the kind of equation you'd see on the GMAT, and it would be more realistic if the quadratic had a simple factorization, so we could see using simple algebra that the answer is sometimes yes, sometimes no. Regardless, it's not sufficient.
When we use Statement 2, we can replace b with -1 to rephrase our question:
"Does x = 9 - (-1)x ?"
"Does x = 9 + x ?"
"Does 0 = 9 ?"
and clearly the answer to that question is 'no', so Statement 2 is sufficient. The answer is B.
