Question

solve the equation in the interval [0,2pi)
2cos^2x+5cosx+2=0

Answer 1

\[2\cos^2(x) +5\cos(x) +2=0\]let \(a = \cos(x)\)\[\color{red}{2}a^2 +5a+\color{red}{2}=0~~ \sf \text{here we can use the BUM method for simplifying this quadratic}\]\[a^2+5a+4=0\]\[(a+4)(a+1)=0\]\[\left(a+\frac{4}{2}\right)\left(a+\frac{1}{2}\right)=0\]\[(a+2)(2a+1)=0\]Resub \(a=\cos(x)\) \[(\cos(x) +2)(2\cos(x)+1)=0\]Now solve for both factors.