If m and n are real numbers, what is the value of m^2 - n^2? (1) (m + n)^2 = 36 (2) m^2 + … — GMAT

If $m$ and $n$ are real numbers, what is the value of $m^2 - n^2$ ?

(1) $(m + n)^2 = 36$
(2) $m^2 + n^2 = 20$

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.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.Cevap

Statements (1) and (2) TOGETHER are NOT sufficient.

The correct response identifies that statements (1) and (2) together leave sign ambiguity in the difference of squares because

m

and

n

can swap values while maintaining the same sum and sum of squares.

Rephrase the target question

The target expression is

m^2 - n^2 = (m + n)(m - n)

. To find a unique value, we need a unique numerical value for

(m + n)(m - n)

Factoring the difference of squares clarifies what information is necessary.

Evaluate Statement (1) alone

(m + n)^2 = 36 \implies m + n = 6

m + n = -6

. We have no information about

m - n

Since

m - n

can take infinitely many values,

m^2 - n^2

is not uniquely determined. Statement (1) is insufficient.

Evaluate Statement (2) alone

m^2 + n^2 = 20

Different pairs of numbers like

(m, n) = (\sqrt{20}, 0)

give

m^2 - n^2 = 20

, while

(m, n) = (0, \sqrt{20})

give

m^2 - n^2 = -20

. Statement (2) is insufficient.

Evaluate Statements (1) and (2) combined

Expand Statement (1):

m^2 + 2mn + n^2 = 36

. Substitute Statement (2):

20 + 2mn = 36 \implies 2mn = 16 \implies mn = 8

.
Now test values satisfying

m + n = 6

and

mn = 8

:
Case 1:

m = 4, n = 2 \implies m^2 - n^2 = 4^2 - 2^2 = 16 - 4 = 12

.
Case 2:

m = 2, n = 4 \implies m^2 - n^2 = 2^2 - 4^2 = 4 - 16 = -12

.
Since

m^2 - n^2

yields two distinct values (

12

and

-12

), the statements together are insufficient.

Symmetry between variables in a non-linear system creates multiple possible values for non-symmetric target expressions.

Symmetry and Degree Ambiguity in Algebraic Data Sufficiency

Soru