If u^2 + (u − 2v − 1)^2 = −4v(u + v), then what is the value of u + 3v

If u2 + (u − 2v − 1)2 = −4v(u + v), then what is the value of u + 3v?

  1. 1/4
  2. 0
  3. 1/2
  4. -1/4

Answer

u2 + (u − 2v − 1)2 = −4v(u + v)

⇒ u2 + u2 + 4v2 + 1 − 4uv + 4v − 2u + 4vu + 4v2 = 0

⇒ 2u2 − 2u + 8v2 + 4v + 1 = 0

⇒ 2(u2 − u + 1/4) + 2(4v2 + 2v + 1/4) = 0

⇒ 2(u − 1/2)2 + 2(2v + 1/2)2 = 0

⇒ u − 1/2 = 0; 2v + 1/2 = 0

u = 1/2 and v = −1/4

u + 3v = 1/2 − 3/4 = −1/4

The correct option is D.