If a, b, c are real and a² + b² + c² = 2(a - b - c) -3, then the value of 2a - 3b + 4c is?

Solution (By Examveda Team)

a² + b² + c² = 2(a - b - c) - 3
⇒ a² + b² + c² = 2a - 2b - 2c - 3
⇒ a² + b² + c² - 2a + 2b +2c + 1 + 1 + 1 = 0
⇒ (a² - 2a + 1) + (b² + 2b + 1) + (c² + 2c + 1) = 0
⇒ (a - 1)² + (b + 1)² + (c + 1)² = 0
a = 1
b = -1
c = -1
∴ 2a - 3b + 4c
= 2 × 1 - 3 × (-1) + 4 × (-1)
= 2 + 3 - 4
= 1