Solve for x 2cos(x)^2-3cos(x)+1=0

Problem

2*cos2(x)−3*cos(x)+1=0

Solution

1. Substitute a variable to simplify the quadratic form by letting u=cos(x)

2*u2−3*u+1=0

2. Factor the quadratic equation by finding two numbers that multiply to 2 and add to −3

(2*u−1)*(u−1)=0

3. Solve for u by setting each factor equal to zero.

2*u−1=0⇒u=1/2

u−1=0⇒u=1

4. Back-substitute cos(x) for u to find the values of x

cos(x)=1/2

cos(x)=1

5. Determine the general solutions for x based on the unit circle.

x=±π/3+2*k*π

x=2*k*π

Final Answer

x=2*k*π,±π/3+2*k*π

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