If x and y are real numbers such that 4x^2 + 4y^2 − 4xy − 6y + 3 = 0, then the value of (4x + 5y) is

Question

UPSC Quizroom · Accepted Answer

We know:
4x^2 + y^2 − 4xy = (2x − y)^2

Also:
3y^2 − 6y + 3 = 3(y^2 − 2y + 1) = 3(y − 1)^2

Therefore:
4x^2 + 4y^2 − 4xy − 6y + 3 = (2x − y)^2 + 3(y − 1)^2 = 0

Since the sum of squares = 0, each term must be zero:
2x − y = 0 → y = 2x
y − 1 = 0 → y = 1

Substitute y = 1:
1 = 2x → x = 1/2

Now calculate:
4x + 5y = 4(1/2) + 5(1) = 2 + 5 = 7

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