If p and q are real numbers, what is the value of p^2 + 4q^2? (1) p + 2q = 8 (2) pq = 6 — GMAT

If $p$ and $q$ are real numbers, what is the value of $p^2 + 4q^2$ ?

(1) $p + 2q = 8$
(2) $pq = 6$

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

BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The target expression

p^2 + 4q^2

can be related to the binomial square

(p + 2q)^2 = p^2 + 4pq + 4q^2

. Rearranging gives

p^2 + 4q^2 = (p + 2q)^2 - 4pq

. Neither statement alone provides both

p + 2q

and

pq

. However, combining Statement (1) (

p + 2q = 8

) and Statement (2) (

pq = 6

) allows direct substitution:

p^2 + 4q^2 = 8^2 - 4(6) = 40

. Because this yields a single unique value, both statements together are sufficient.

Evaluate Statement (1) independently

Statement (1) states

p + 2q = 8

. Squaring both sides yields

(p + 2q)^2 = p^2 + 4pq + 4q^2 = 64

, so

p^2 + 4q^2 = 64 - 4pq

. Without the value of

pq

, a unique numerical value for

p^2 + 4q^2

cannot be determined.

Statement (1) alone leaves one degree of freedom (the product

pq

is unknown).

Evaluate Statement (2) independently

Statement (2) states

pq = 6

. Knowing only the product of

p

and

q

allows infinitely many pairs

(p, q)

, resulting in infinitely many values for

p^2 + 4q^2

Statement (2) alone does not constrain the linear sum

p + 2q

Combine Statements (1) and (2)

From Statement (1),

p^2 + 4q^2 = 64 - 4pq

. Substituting

pq = 6

from Statement (2) yields

p^2 + 4q^2 = 64 - 4(6) = 40

. This provides a unique, definitive numerical answer.

Combining both statements eliminates all unknown parameters from the target expression

p^2 + 4q^2

Algebraic Identity Expansion and System Combination in Data Sufficiency

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