# If alpha is a root of 4x^2+2x-1 =0 . prove that 4(alpha)^3- 3(alpha) is the other root.

Solution:   Let other root is $\beta$ , then

$\alpha+\beta =-\frac{2}{4}=-\frac{1}{2}\Rightarrow \beta=-\frac{1}{2}-\alpha$

and  so , $4(\alpha)^{2}+2(\alpha)-1=0,$  because $\alpha$ is a root $4x^{2}+2x-1=0$

Now ,                   $\beta=4(\alpha)^{3}-3(\alpha)=\alpha(1-2\alpha-3)=-\frac{1}{2}-\alpha$

Hence   ,  $4(\alpha)^{3}-3(\alpha)$  is the other root.

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