`2cos^2 (x) + cos(x) - 1 = 0` Find all the solutions of the equation in the interval `0,2pi)`.

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Chapter 5, 5.3 - Problem 33 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Find all solutions to the equation `2cos^2(x)+cos(x)-1=0` in the interval `[0,2pi).`

`2cos^2(x)+cos(x)-1=0`

`(2cos(x)-1)(cos(x)+1)=0`

Set each factor equal to zero and solve for the x value(s).

`2cos(x)-1=0`

`2cos(x)=1`

`cos(x)=1/2`

`x=pi/3,x=(5pi)/3`

`cos(x)+1=0`

`cos(x)=-1`

`x=pi` 

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