Q13/126) If one of the roots of the equation 2x^2 - x -2 = 0 is a, prove that the other root is 4a^3 - 6a - 3/2

One solution is a so

2a^2 –a -2 = 0

Let other root be p = 4a^3 -6a – 3/2

So 4a^3 – 4 a = 2a^2

So 4a^3 -6a – 3/2 = 2a^2  – 2a – 3/2 = (2a^2 – a – 2) – a + ½ = ½ -- a

Now sum of roots = ½

Product of roots =  a(1/2- a) = ½a(1-2a) = 1/a – a^2 = - ½(2a^2- a)  = -1

So p is another root