Nothing to do with Cardano's

Algebra Level 3

If a , b a,b and c c are the roots to the equation x 3 + x 2 + x − 1 = 0 x^3 + x^2 + x - 1= 0 , then which of the following equations have roots a 2 , b 2 a^2,b^2 and c 2 c^2 ?

− x 3 + x 2 + 3 x + 1 = 0 -x^3 + x^2 +3x + 1 = 0 x 3 + x 2 + 3 x − 1 = 0 x^3 + x^2 + 3x - 1 = 0 x 3 − x 2 + 3 x + 1 = 0 x^3 -x^2 +3x +1 = 0 x 3 + x 2 − 3 x + 1 = 0 x^3 + x^2 -3x + 1 = 0

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Rajen Kapur
Aug 5, 2016

x 2 − 1 = − x ( x 2 + 1 ) x^2-1=-x(x^2+1) Squaring and replacing x 2 x^2 by t, we get ( t − 1 ) 2 = t ( t + 1 ) 2 → t 2 − 2 t + 1 = t 3 + 2 t 2 + t (t-1)^2=t(t+1)^2 \rightarrow t^2-2t+1=t^3+2t^2+t . Rearranging we get the equation with squared roots x 3 + x 2 + 3 x − 1 = 0 x^3+x^2 +3x - 1=0

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