`tan^2 (x) + 3tan(x) + 1 = 0` Use the Quadratic Formula to solve the equation in the interval `0,2pi)`. Then use a graphing utility to approximate the angle `x`.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`tan^2(x)+3tan(x)+1=0`

using quadratic equation formula,

`tan(x)=(-3+-sqrt(3^2-4*1*1))/2`

`tan(x)=(-3+-sqrt(9-4))/2`

`tan(x)=(-3+-sqrt(5))/2`

Solutions for `tan(x)=(-3-sqrt(5))/2`  for the range `0<=x<=2pi`  are,

`x=pi+arctan((-3-sqrt(5))/2) , x=2pi+arctan((-3-sqrt(5))/2)`

Solutions for `tan(x)=(-3+sqrt(5))/2`  for the range `0<=x<=2pi`  are,

`x=pi+arctan((-3+sqrt(5)/2) , x=2pi+arctan((-3+sqrt(5))/2)`

See the attached graph

Approximate x= 1.9 , 2.8 , 5.1 , 5.9

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