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Difficulty: Very hardAlgebraic Equations and Systems in Data Sufficiency

If aa and bb are real numbers, what is the value of (a+b)2(a + b)^2?

(1) a2+b2+4a−6b=−13a^2 + b^2 + 4a - 6b = -13
(2) ab=−6ab = -6

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  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. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) can be rewritten by completing the square as (a+2)2+(b−3)2=0(a + 2)^2 + (b - 3)^2 = 0. Since aa and bb are constrained to be real numbers, (a+2)2≥0(a + 2)^2 \geq 0 and (b−3)2≥0(b - 3)^2 \geq 0. The only way their sum can equal 00 is if a=−2a = -2 and b=3b = 3. This uniquely determines (a+b)2=(−2+3)2=1(a + b)^2 = (-2 + 3)^2 = 1, making Statement (1) alone sufficient. Statement (2) gives ab=−6ab = -6, which allows multiple values for (a+b)2(a + b)^2 (e.g., 11 or 2525), making it insufficient. Hence, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not is the correct choice.

Step-by-Step Solution

1
Analyze Statement (1): a2+b2+4a−6b=−13a^2 + b^2 + 4a - 6b = -13
Rearrange and group terms: (a2+4a)+(b2−6b)=−13(a^2 + 4a) + (b^2 - 6b) = -13. Complete the square for both variables by adding 44 and 99 to both sides: (a2+4a+4)+(b2−6b+9)=−13+4+9(a^2 + 4a + 4) + (b^2 - 6b + 9) = -13 + 4 + 9, which simplifies to (a+2)2+(b−3)2=0(a + 2)^2 + (b - 3)^2 = 0.
Grouping and completing the square reveals the sum-of-squares structure.
2
Evaluate the real number constraint on Statement (1)
Because aa and bb are real numbers, (a+2)2≥0(a + 2)^2 \geq 0 and (b−3)2≥0(b - 3)^2 \geq 0. The sum of two non-negative terms can equal zero if and only if both terms are independently zero: a+2=0  ⟹  a=−2a + 2 = 0 \implies a = -2 and b−3=0  ⟹  b=3b - 3 = 0 \implies b = 3.
The sum of non-negative real squares equaling zero forces each squared term to be zero.
3
Calculate the target expression using values from Statement (1)
(a+b)2=(−2+3)2=(1)2=1(a + b)^2 = (-2 + 3)^2 = (1)^2 = 1. Since this yields a single unique value, Statement (1) ALONE is sufficient.
A unique value for the target expression establishes sufficiency.
4
Analyze Statement (2): ab=−6ab = -6
If a=2a = 2 and b=−3b = -3, then ab=−6ab = -6 and (a+b)2=(2−3)2=1(a + b)^2 = (2 - 3)^2 = 1. If a=1a = 1 and b=−6b = -6, then ab=−6ab = -6 and (a+b)2=(1−6)2=25(a + b)^2 = (1 - 6)^2 = 25. Multiple values are possible.
Testing specific valid cases proves that Statement (2) does not yield a unique result.
5
Conclude Data Sufficiency determination
Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.
Statement (1) produces a unique answer, whereas Statement (2) does not.

Key Concept

Completing the Square and Trivial Inequality for Real Squares in Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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