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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If aa and bb are real numbers, what is the value of a2−b2a^2 - b^2?

(1) a+b=5a + b = 5
(2) (a−b)2=9(a - b)^2 = 9

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.Answer

Answer

The statements together are not sufficient to determine a unique value for a2−b2a^2 - b^2, because a2−b2a^2 - b^2 can equal either 1515 or −15-15.
The correct option states that both statements together are not sufficient. Factoring the target expression yields a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). Statement (1) tells us a+b=5a+b = 5, and Statement (2) tells us a−b=±3a-b = \pm 3. Combining them yields two possible numerical results (1515 and −15-15). Since Data Sufficiency requires a single, unique numerical value, the information remains insufficient.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities.
The target expression a2−b2a^2 - b^2 factors into (a+b)(a−b)(a + b)(a - b). To find a unique value, we need a unique value for the product (a+b)(a−b)(a + b)(a - b).
Factoring highlights the required components: the sum (a+b)(a + b) and the difference (a−b)(a - b).
2
Evaluate Statement (1) independently.
Statement (1) gives a+b=5a + b = 5. However, the value of a−ba - b is completely unknown.
Since (a−b)(a - b) can be any real number, a2−b2=5(a−b)a^2 - b^2 = 5(a - b) can take infinitely many values. Statement (1) is not sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives (a−b)2=9(a - b)^2 = 9, which implies a−b=3a - b = 3 or a−b=−3a - b = -3.
The sum a+ba + b is completely unknown, and a−ba - b has two potential values. Statement (2) is not sufficient.
4
Evaluate Statements (1) and (2) together.
From (1), a+b=5a + b = 5. From (2), a−b=3a - b = 3 or a−b=−3a - b = -3.
If a−b=3a - b = 3, then a2−b2=(5)(3)=15a^2 - b^2 = (5)(3) = 15.
If a−b=−3a - b = -3, then a2−b2=(5)(−3)=−15a^2 - b^2 = (5)(-3) = -15.
Because there are two distinct outcomes (1515 and −15-15), a unique value cannot be determined. Therefore, both statements together are not sufficient.

Key Concept

Quadratic non-linearity and root ambiguity in Data Sufficiency systems
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Topics

Algebraic Equations and Systems in Data Sufficiency

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