If x and y are real numbers, what is the value of x + y? (1) x^2 + 2xy + y^2 - 5x - 5y + 6… — GMAT

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $x^2 + 2xy + y^2 - 5x - 5y + 6 = 0$
(2) $xy = 2$

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The correct response identifies that neither statement alone establishes a single value for

x + y

, but combining them eliminates

x + y = 2

because no real numbers

x

and

y

have a sum of

2

and a product of

2

(discriminant

2^2 - 4(2) = -4 < 0

). Only

x + y = 3

yields real values for

x

and

y

(discriminant

3^2 - 4(2) = 1 \ge 0

). Thus, both statements together are sufficient to determine that

x + y = 3

Step-by-Step Solution

Analyze Statement (1) by grouping terms in terms of

(x + y)

Rewrite

x^2 + 2xy + y^2 - 5(x + y) + 6 = 0

(x + y)^2 - 5(x + y) + 6 = 0

, which factors into

((x + y) - 2)((x + y) - 3) = 0

. Hence,

x + y = 2

x + y = 3

. Since there are two possible values, Statement (1) alone is NOT sufficient.

Determining whether Statement (1) provides a unique value for the target expression.

Analyze Statement (2) independently.

Statement (2) gives

xy = 2

. Multiple real pairs satisfy this (e.g.,

(2, 1) \Rightarrow x + y = 3

(1, 2) \Rightarrow x + y = 3

vs.

(\sqrt{2}, \sqrt{2}) \Rightarrow x + y = 2\sqrt{2}

). Thus, Statement (2) alone is NOT sufficient.

Testing whether

xy = 2

alone restricts

x + y

to a single numerical value.

Combine Statement (1) and Statement (2) under the real number constraint.

Let

S = x + y

and

P = xy = 2

. For

x

and

y

to be real numbers, they must be the real roots of

t^2 - St + P = 0

, requiring discriminant

\Delta = S^2 - 4P \ge 0

.
- Case 1: If

S = 2

, then

\Delta = 2^2 - 4(2) = -4 < 0

. No real solutions exist for

x

and

y

.
- Case 2: If

S = 3

, then

\Delta = 3^2 - 4(2) = 1 \ge 0

. Real solutions exist (e.g.,

x = 2, y = 1

Applying the real-number constraint to eliminate algebraically impossible cases.

Conclude total sufficiency.

Only

x + y = 3

produces real values for

x

and

y

. Therefore, both statements together uniquely determine

x + y = 3

Completing the Data Sufficiency evaluation.

Key Concept

Quadratic expression rephrasing and discriminant constraints for real variable existence in Data Sufficiency

Estimated Time:2m 0s

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