If `alpha` is a root of the equation `4x^2+2x-1=0,` then prove that `4alpha^3-3alpha` is the other root.

Explanation

We have `4x^(2) + 2x - 1 = 0`
Let the other root be `beta` .
`therefore alpha + beta = - (1)/(2), alpha beta = - (1)/(4)`
Also, `4alpha^(2) + 2alpha - 1 = 0 as alpha` is a root, and we have to prove that
`beta = 4alpha^(3) - 3 alpha.` Now
`4alpha^(3) - 3alpha = 4 alpha ^(2) alpha - 3 alpha`
= ` alpha (1 - 2 alpha) - 3 alpha `
`= - (1)/(2)[4alpha^(2) - 2alpha]`
=`-(1)/(2)[1-2alpha + 4 alpha]`
`= - (1)/(2) (1 + 2 alpha ) = - (1)/(2) - alpha = beta` [From (1)]
Hence, the other root `beta is 4alpha^(3) - 3 alpha`.