Solve a quadratic ratio identity: If P^2 + 4Q^2 = 4PQ, determine the ratio P : Q in lowest terms.

Introduction / Context:
Algebraic identities often hide a perfect square. Recognizing and factoring them lets you deduce a fixed ratio between variables. Here, the expression rearranges into a perfect square in P and Q.

Given Data / Assumptions:

• P^2 + 4Q^2 = 4PQ.
• P and Q are real numbers (nontrivial case).

Concept / Approach:
Bring all terms to one side and try to factor. If the left-hand side becomes a perfect square, set the factor to zero to find a linear relation between P and Q, which yields the ratio.

Step-by-Step Solution:
P^2 − 4PQ + 4Q^2 = 0.This is (P − 2Q)^2 = 0.Therefore P − 2Q = 0 ⇒ P = 2Q.Hence P : Q = 2 : 1.

Verification / Alternative check:
Substitute P = 2Q into the original: LHS = (2Q)^2 + 4Q^2 = 4Q^2 + 4Q^2 = 8Q^2; RHS = 4 * (2Q) * Q = 8Q^2, identity holds.

Why Other Options Are Wrong:

• 1 : 3, 3 : 1, 1 : 2 do not satisfy the given identity when substituted.

Common Pitfalls:

• Missing the perfect-square pattern and attempting unnecessary quadratic solutions.

Final Answer:
2 : 1