solve the equation for x in the interval 02pi Use exact solu

solve the equation for x in the interval [0,2pi). Use exact solutions where possible and give approximate solutions correct to four decimal places
3tan^2x+5tanx+2=0

Solution

3tan^2x+5tanx+2=0 interval [0,2pi)

Factorise : 3tan^2x+5tanx+2

3tan^2x +3tanx +2tanx +2 =0

3tanx( tanx +1) +2(tanx +1) =0

(3tanx +2)( tanx +1) =0

3tanx +2 =0

tanx = -2/3

x = pi - 0.5880 = 2.5535 rad

x = 2pi - 0.5880 = 5.6951 rad

tanx +1 =0

x = pi -pi/4 , 2pi -pi/4

= 3pi/4 , 7pi/4 = 2.3561 , 5.4977

Solution : x = 2.3561  , 2.5535 , 5.4977 , 5.6951,