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Difficulty: EasyAlgebraic Equations and Systems in Data Sufficiency

If xx is a real number, what is the value of (x+3)2(x + 3)^2?

(1) x2+6x=7x^2 + 6x = 7
(2) x=1x = 1

  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. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Each statement alone is sufficient.
Expanding the target expression (x+3)2(x + 3)^2 gives x2+6x+9x^2 + 6x + 9. Statement (1) provides x2+6x=7x^2 + 6x = 7, which can be directly substituted into the expanded expression to get 7+9=167 + 9 = 16, yielding a unique value. Statement (2) explicitly provides x=1x = 1, which gives (1+3)2=16(1 + 3)^2 = 16. Because each statement independently provides enough information to determine a single value for the expression, the option stating that each statement alone is sufficient is correct.

Step-by-Step Solution

1
Rephrase the target expression using algebraic expansion.
(x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9
Expanding the binomial allows direct comparison with given algebraic statements.
2
Evaluate Statement (1) using the expanded target expression.
Since x2+6x=7x^2 + 6x = 7, substituting this into x2+6x+9x^2 + 6x + 9 gives 7+9=167 + 9 = 16. Alternatively, solving x2+6x−7=0x^2 + 6x - 7 = 0 yields x=1x = 1 or x=−7x = -7. If x=1x = 1, (1+3)2=16(1 + 3)^2 = 16. If x=−7x = -7, (−7+3)2=(−4)2=16(-7 + 3)^2 = (-4)^2 = 16. Both roots yield the exact same value of 1616.
Statement (1) alone uniquely determines the target value.
3
Evaluate Statement (2) independently.
Given x=1x = 1, substituting directly into the target expression gives (1+3)2=42=16(1 + 3)^2 = 4^2 = 16.
Statement (2) alone uniquely determines the target value.
4
Conclude sufficiency based on independent evaluations.
Both Statement (1) alone and Statement (2) alone are sufficient.
Since each statement independently determines a single value of 16, EACH statement ALONE is sufficient.

Key Concept

Algebraic Expression Expansion and Target Rephrasing in Data Sufficiency
Estimated Time:1m 0s
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Topics

Algebraic Equations and Systems in Data Sufficiency

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