Find all Solutions

Question

Find all solutions to the equation 2 cos^2 x -3 cos x +1 =0 Let k be any integer.

Answer 1

Let w = cos(x)

Then 2cos²(x) - 3cos(x) + 1 = 0 turns into 2w² - 3w + 1 = 0 and you can factor that into (2w - 1)(w - 1) = 0 and solve giving you w = 1/2 or w = 1. Now back-substitute to get cos(x) = 1/2 or cos(x) = 1 and use the unit circle to solve. x = π/3, x = 5π/3, x = 0.

Since there are no domain restrictions you'd need to make the answers generic by adding 2πk to each:

x = π/3 + 2πk, x = 5π/3 + 2πk, x = 2πk