MathLabs

Problem 3

Let a,b,ca,b,c be real numbers. Consider acos⁡2x+bcos⁡x+c=0a\cos^2x+b\cos x+c=0. Using a,b,ca,b,c, form a quadratic equation in cos⁡2x\cos2x whose roots are the same values of xx. Compare the equations for a=4a=4, b=2b=2, c=−1c=-1.
Step 4 of 4: Compare the numerical instance
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4cos⁡2x+2cos⁡x−1=0↔4cos⁡22x+2cos⁡2x−1=04\cos^2x+2\cos x-1=0\quad\leftrightarrow\quad4\cos^22x+2\cos2x-1=0
Unit circle showing the angle 72∘72^\circ
The marked angle has cosine as its horizontal coordinate; doubling the angle links it to the companion root.
Detailed analysis

For a=4a=4, b=2b=2, c=−1c=-1, the original equation is 4cos⁡2x+2cos⁡x−1=04\cos^2x+2\cos x-1=0. Substitution into the general formula gives 4cos⁡22x+2cos⁡2x−1=04\cos^22x+2\cos2x-1=0, so the two equations have the same xx-solutions.