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 3 of 4: Use the double-angle variable
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a2z2+(2a2+4ac−2b2)z+(a2+4ac−2b2+4c2)=0,z=cos⁡2xa^2z^2+(2a^2+4ac-2b^2)z+(a^2+4ac-2b^2+4c^2)=0,\quad z=\cos2x
Detailed analysis

Set z=cos⁡2xz=\cos2x, so cos⁡2x=(z+1)/2\cos^2x=(z+1)/2. Substitution and collection of terms yields the required quadratic in zz.