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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

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

(1) a4−b4=5(a2−b2)a^4 - b^4 = 5(a^2 - b^2)
(2) ab=6ab = 6 and a+b=5a + b = 5

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) simplifies to (a2−b2)(a2+b2−5)=0(a^2 - b^2)(a^2 + b^2 - 5) = 0, which allows a2+b2=5a^2 + b^2 = 5 when a≠ba \neq b, but allows infinitely many values for a2+b2a^2 + b^2 when a=ba = b. Hence, Statement (1) alone is insufficient. Statement (2) provides a+b=5a + b = 5 and ab=6ab = 6, allowing us to use (a+b)2=a2+b2+2ab(a + b)^2 = a^2 + b^2 + 2ab to find a2+b2=25−12=13a^2 + b^2 = 25 - 12 = 13 uniquely. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1
Analyze Statement (1) algebraically.
Factor the left side as (a2−b2)(a2+b2)=5(a2−b2)(a^2 - b^2)(a^2 + b^2) = 5(a^2 - b^2). Rearranging yields (a2−b2)(a2+b2−5)=0(a^2 - b^2)(a^2 + b^2 - 5) = 0.
Dividing by (a2−b2)(a^2 - b^2) is valid only if a≠ba \neq b.
2
Evaluate sufficiency of Statement (1).
If a≠ba \neq b, then a2+b2=5a^2 + b^2 = 5. However, if a=b>0a = b > 0, then a2−b2=0a^2 - b^2 = 0, which makes the equation true for any positive value of aa, meaning a2+b2=2a2a^2 + b^2 = 2a^2 can take infinitely many values (e.g., if a=b=1a=b=1, a2+b2=2a^2+b^2=2; if a=b=3a=b=3, a2+b2=18a^2+b^2=18). Thus, a unique value cannot be determined.
Multiple possible values mean Statement (1) is not sufficient.
3
Evaluate sufficiency of Statement (2).
Use the algebraic identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2. Substituting a+b=5a + b = 5 and ab=6ab = 6 gives 52=a2+2(6)+b25^2 = a^2 + 2(6) + b^2, so 25=a2+b2+12  ⟹  a2+b2=1325 = a^2 + b^2 + 12 \implies a^2 + b^2 = 13.
This determines a single, unique numerical value for a2+b2a^2 + b^2.
4
Conclude overall sufficiency.
Statement (1) is NOT sufficient, but Statement (2) IS sufficient.
Statement (2) alone resolves the target question.

Key Concept

Algebraic Equations and Systems in Data Sufficiency
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Algebraic Equations and Systems in Data Sufficiency

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