Math Olympiad | A Nice Exponential Problem | VIJAY Maths

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[prime]

One can guess an answer immediately. Inspiration beats perspreation!!

[RealQinnMalloryu]

{36x^4+44x^4}= 80x^8 8^10^x^2^3 2^32^5x^2^3 1^3^1^1x2^1 32 (x ➖ 3x+2)

[andryvokubadra]

706 = 3⁴ + 5⁴ = 5⁴ + 3⁴
(x+9)⁴ + (x+11)⁴ = 3⁴ + 5⁴
(x+9)⁴ + (x+11)⁴ = 5⁴ + 3⁴


(x+9)⁴ = 3⁴
x+9 = ±3
x + 9 = 3 --> x = -6
x + 9 = -3 --> x = -12

(x+9)⁴ = 5⁴
x+9 = ±5
x + 9 = 5 --> x = -4
x + 9 = -5 --> x = -14

(x+11)⁴ = 5⁴
x+11 = ±5
x+11 = 5 --> x = -6
x+11 = -5 --> x = -16

(x+11)⁴ = 3⁴
x+11 = ±3
x+11 = 3 --> x = -8
x+11 = -5 --> x = -16


Verified :
(x+9)⁴ + (x+11)⁴ = 706

x = -4 --> (5)⁴ + (7)⁴ > 706 (Unmatch)
x = -6 --> (3)⁴ + (5)⁴ = 706 (Match)
x = -8 --> (1)⁴ + (3)⁴ < 706 (Unmatch)
x = -12 --> (-3)⁴ + (-1)⁴ < 706 (Unmatch)
x = -14 --> (-5)⁴ + (-3)⁴ = 706 (Match)
x = -16 --> (-7)⁴ + (-5)⁴ > 706 (Unmatch)


So x = -6, -14 11:20