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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If pp and qq are real numbers, what is the value of (p+q)2(p + q)^2?

(1) p2+q2=25p^2 + q^2 = 25
(2) pq=12pq = 12

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Expanding the target expression gives (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2. Statement (1) provides p2+q2=25p^2 + q^2 = 25 but leaves pqpq unknown, making it insufficient alone. Statement (2) provides pq=12pq = 12 but leaves p2+q2p^2 + q^2 unknown, making it insufficient alone. Combining both statements allows direct substitution into the identity: (p+q)2=25+2(12)=49(p + q)^2 = 25 + 2(12) = 49, which yields a single unique answer.

Step-by-Step Solution

1
Rephrase the question stem algebraically.
(p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2
Expanding the target expression shows that calculating (p+q)2(p + q)^2 requires knowing the sum of squares (p2+q2)(p^2 + q^2) and the product pqpq.
2
Evaluate Statement (1) independently.
Insufficient
Given p2+q2=25p^2 + q^2 = 25, the term 2pq2pq remains unknown. For example, if p=5p = 5 and q=0q = 0, then (p+q)2=25(p+q)^2 = 25; if p=3p = 3 and q=4q = 4, then (p+q)2=49(p+q)^2 = 49. Since multiple outcomes exist, Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Insufficient
Given pq=12pq = 12, the term p2+q2p^2 + q^2 remains unknown. For example, if p=3p = 3 and q=4q = 4, then (p+q)2=49(p+q)^2 = 49; if p=1p = 1 and q=12q = 12, then (p+q)2=169(p+q)^2 = 169. Since multiple outcomes exist, Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together.
Sufficient
Substituting p2+q2=25p^2 + q^2 = 25 and pq=12pq = 12 into (p+q)2=(p2+q2)+2(pq)(p + q)^2 = (p^2 + q^2) + 2(pq) yields 25+2(12)=4925 + 2(12) = 49. This determines a unique value.

Key Concept

Algebraic Identity Expansion and Substitution in Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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