How to solve this nice math Exponential algebra problem | Olympiad Question | x=?

[@AfsarSaid-f8s]

Great question

[@HuzaifaKhan-m3i]

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[@miguelgnievesl6882]

I started by factoring x⁴-6x³+9x²+100=(x²+ax+5)(x²+bx+20). I got a=2 and b=-8. So x⁴-6x³+9x²+100=(x²+2x+5)(x²-8x+20). I applied Pythagoras to both factors and the result is: x=-1+2i; x=-1-2i; x=4+2i; x=4-2i

[@raghvendrasingh1289]

👍 Second method (x^2 - 3x)^2 = -100 (x^2 - 3x+k)^2 = 2 kx^2-6 kx+(k^2-100) for RHS to be perfect square 4 ac = b^2 8 k(k^2-100) = 36 k^2 k (2 k^2-9 k -200) = 0 k(2 k -25)(k+8)= 0 we want coefficient of x^2 in RHS positive hence k = 25/2 (x^2-3x+25/2)^2 = (5x - 15/2)^2 (x^2+2x+5)(x^2 - 8 x +20) = 0 x = -1+2 i , -1- 2 i , 4+2 i , 4-2 i