3y2 + 7y + 1 = 0

(vii) 3y² + 7y + 1 = 0 

Sol. 3y² + 7y + 1 = 0

∴ 3y² + 7y = –1

Dividing both the sides by 3 we get,

y² + (⁷/₃ )(y) = ^{- 1} / ₃      .....(i)

Third term = ( ½ ×  coefficient of y)²

= ( ½ × ⁷/₃)²

= (⁷/₆)²

= ⁴⁹/₃₆

Adding ⁴⁹/₃₆  to both sides of (i) we get,

y² + (⁷/₃ )(y) + ⁴⁹/₃₆   = ^{- 1} / ₃   + ⁴⁹/₃₆  

∴ (y + ⁷/₆)² = ³⁷/₃₆

∴  y + ⁷/₆ = ± √(³⁷/₃₆)

∴  y + ⁷/₆ = ± √³⁷/₆

∴ y = -  ⁷/₆ ± √³⁷/₆

∴ y = ^{(- 7 ± √37)}/₆

∴ y = ^{(- 7 + √37)}/₆  or  ^{(- 7 -  √37)}/₆