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

Problem

2*cos2(x)+3*cos(x)−2=0

Solution

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

2*u2+3*u−2=0

2. Factor the quadratic equation by finding two numbers that multiply to −4 and add to 3

(2*u−1)*(u+2)=0

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

2*u−1=0⇒u=1/2

u+2=0⇒u=−2

4. Back-substitute u=cos(x) into the equations.

cos(x)=1/2

cos(x)=−2

5. Evaluate the solutions for x Since the range of the cosine function is [−1,1] the equation cos(x)=−2 has no real solutions.

cos(x)=−2⇒No solution

6. Determine the general solutions for cos(x)=1/2 using the unit circle.

x=π/3+2*n*π

x=−π/3+2*n*π

Final Answer

x=±π/3+2*n*π

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