If one root of the equation $4x^2 + 2x - 1 = 0$ is $\alpha$, then other root is -

Multiplying the equation by 2 gives 8x^3 + 4x^2 - 2x = 0, which can be factored as 2x(4x^2 + 2x - 1) = 0. This shows the roots of the original quadratic 4x^2 + 2x - 1 = 0 are also roots of the cubic equation 4x^3 + 2x^2 - x = 0, alongside x = 0. Substituting x = 2*alpha into the cubic equation yields 4(2*alpha)^3 + 2(2*alpha)^2 - (2*alpha) = 0, which simplifies to 32*alpha^3 + 8*alpha^2 - 2*alpha = 0. Dividing by 2*alpha gives 16*alpha^2 + 4*alpha - 1 = 0, and since alpha is a root of the original equation, we know 4*alpha^2 + 2*alpha - 1 = 0. Adding these two results together confirms that 2*alpha is the other root.

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