Solve for x 2cos(x)^2-7cos(x)+3=0

Problem

2*cos2(x)−7*cos(x)+3=0

Solution

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

2*u2−7*u+3=0

2. Factor the quadratic equation by finding two numbers that multiply to 2⋅3=6 and add to −7

(2*u−1)*(u−3)=0

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

2*u−1=0⇒u=1/2

u−3=0⇒u=3

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

cos(x)=1/2

cos(x)=3

5. Evaluate the solutions for x Since the range of the cosine function is [−1,1] the equation cos(x)=3 has no real solutions.

cos(x)=3⇒No solution

6. Determine the general solution 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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