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 2 of 4: Square and remove the odd power
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4a2cos⁡4x+2(4ac−2b2)cos⁡2x+4c2=04a^2\cos^4x+2(4ac-2b^2)\cos^2x+4c^2=0
Detailed analysis

Squaring the preceding identity and multiplying by 44 gives 4a2cos⁡4x+2(4ac−2b2)cos⁡2x+4c2=04a^2\cos^4x+2(4ac-2b^2)\cos^2x+4c^2=0.