Solve by Factoring 2cos(x)^2-cos(x)-1=0

Problem

2*cos2(x)−cos(x)−1=0

Solution

1. Identify the equation as a quadratic in form by letting u=cos(x)

2. Rewrite the equation in terms of u to make factoring easier.

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. Substitute cos(x) back in for u

(2*cos(x)+1)*(cos(x)−1)=0

5. Set each factor equal to zero to solve for cos(x)

2*cos(x)+1=0⇒cos(x)=−1/2

cos(x)−1=0⇒cos(x)=1

6. Solve for x using the unit circle values.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

x=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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