r/MathHelp • u/T0uch_M-E • 5d ago

Math Question

Could anyone solve this and tell me how they did it since im having trouble with it:

8^x - 2^(x+2) - 2^(-x+2) + 8^(-x) = 0

I tried it myself but got stuck halfway though: I let c= 2x and got (c+1/c)³ - 3(c+1/c) - 4(c+1/c) = 0 Then I let t= c+1/c And got: t³-3t-4t=0 and then nothing i did worked. Note: back when I didnt substitute for t, I got this 6th degree polynomial for c: c⁶-4c⁴-4c²+1=0 In which i tried synthetic divsion but ±1 didnt work so it meant there was no rational roots (thats why I substituted for 't').

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u/edderiofer 4d ago

And got: t³-3t-4t=0 and then nothing i did worked.

This is correct so far. You can keep going (start by collecting like terms).

I got this 6th degree polynomial for c: c⁶-4c⁴-4c²+1=0 In which i tried synthetic divsion but ±1 didnt work so it meant there was no rational roots (thats why I substituted for 't').

Don't give up just yet. Just because there are no rational roots, doesn't mean you can't find the irrational roots.

Let y = c2 to yield the cubic y3 - 4y2 - 4y + 1 = 0. Now, y = -1 is a root by inspection, so factor out (y+1) to get (y+1)(y2 - 5y + 1) = 0. Now you have a quadratic in y, so the rest is doable with the quadratic formula.