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

Problem

2*cos2(x)−cos(x)−1=0

Solution

1. Identify the equation as a quadratic in terms of cos(x)

2. Substitute u=cos(x) to rewrite the equation as a standard quadratic.

2*u2−u−1=0

3. Factor the quadratic expression by finding two numbers that multiply to −2 and add to −1

(2*u+1)*(u−1)=0

4. Solve for u by setting each factor to zero.

2*u+1=0⇒u=−1/2

u−1=0⇒u=1

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

cos(x)=−1/2

cos(x)=1

6. Determine the general solutions for x based on the unit circle.
   For cos(x)=1

x=2*n*π

For cos(x)=−1/2

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

x=2*n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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