If a and b are constants, what is the value of a ?

[BTGModeratorVI]

If a and b are constants, what is the value of a ?

(1) a < b
(2) (t − a)( t − b) = t² + t − 12, for all values of t.

Answer: C
Source: Official guide

[Brent@GMATPrepNow]

If a and b are constants, what is the value of a ?

(1) a < b
(2) (t − a)( t − b) = t² + t − 12, for all values of t.

Answer: C
Source: Official guide

Target question: What is the value of a?

Statement 1:a < b
Definitely NOT SUFFICIENT

Statement 2: (t − a)( t − b) = t² + t − 12
Factor: t² + t − 12 = (t + 4)(t - 3)
Rewrite in terms of (t - a) and (t - b) to get: t² + t − 12 = (t - -4)(t - 3)
There are two possible cases:
Case a: a = -4 and b = 3, in which case a = -4
Case b: a = 3 and b = -4, in which case a = 3
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 2 tells us that EITHER a = -4 and b = 3 OR a = 3 and b = -4
Statement 2 tells us that a < b, which means it MUST be the case that a = -4 and b = 3
So, a = -4
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

[Scott@TargetTestPrep]

If a and b are constants, what is the value of a ?

(1) a < b
(2) (t − a)( t − b) = t² + t − 12, for all values of t.

Answer: C
Source: Official guide

Solution:

We need to determine the value of a.

Statement One Alone:

Knowing a < b does not allow us to determine the value of a. Statement one alone is not sufficient.

Statement Two Alone:

Expanding the left side of the equation, we have:

t^2 - (a + b)t + ab = t^2 + t - 12

Equating like terms from both sides, we have:

-(a + b) = 1 and ab = -12

a + b = -1 and ab = -12

The two numbers that add up to -1 and multiply to -12 are -4 and 3. However, it could be the case that a = -4 and b = 3 or the case that a = 3 and b = -4. Statement two alone is not sufficient.

Statements One and Two Together:

With the two statements, we see that a must be -4 and b must be 3 since a < b. Both statements together are sufficient.

Answer: C