Math - Algebra

Linear, quadratic, module, parametric equations

Math - Algebra

I can't solve this equation...
$$(x + 2)^{4}$$ + $$(x + 4)^{4}$$ = 82

Very intresting for me!
Thanks.
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Re: Math - Algebra

You can simplify this a little by letting y= x+ 4 so that x+ 2= y- 2 and then $$(x+ 2)^4+ (x+ 4)^4= (y- 2)^4+ y^4=$$$$y^4- 8y^3+ 24y^2- 32x+ 16+ y^4= 2y^4- 8y^3+ 24y^2- 32y+ 16= 82$$. Subtracting 82 from both sides, $$2y^4- 8y^3+ 24y^2- 32y+ 66= 0$$. Divide by 2: $$y^4- 4y^3+ 12y^2- 16y+ 33= 0$$. By the "rational root theorem" the only possible rational roots are 1, -1, 33, and -33. Evaluating the polynomial at those values shows that -33 is the only rational number root.
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