If alpha is one root of the equation 4x^2 + 2 | vedclass

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(B) Let $\alpha$ and $\beta$ be the roots of the given equation.From the relation between roots and coefficients,the sum of roots is $\alpha + \beta =

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$4\alpha^3 - 3\alpha$

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(B) Let $\alpha$ and $\beta$ be the roots of the given equation.From the relation between roots and coefficients,the sum of roots is $\alpha + \beta = -\frac{2}{4} = -\frac{1}{2}$.Thus,$\beta = -\frac{1}{2} - \alpha$.Since $\alpha$ is a root,$4\alpha^2 + 2\alpha - 1 = 0$,which implies $4\alpha^2 = 1 - 2\alpha$.Now,consider $4\alpha^3 - 3\alpha = \alpha(4\alpha^2) - 3\alpha$.Substituting $4\alpha^2 = 1 - 2\alpha$,we get $\alpha(1 - 2\alpha) - 3\alpha = \alpha - 2\alpha^2 - 3\alpha = -2\alpha^2 - 2\alpha$.From $4\alpha^2 + 2\alpha - 1 = 0$,we have $2\alpha^2 = \frac{1 - 2\alpha}{2} = \frac{1}{2} - \alpha$.Substituting this,we get $-(\frac{1}{2} - \alpha) - 2\alpha = -\frac{1}{2} + \alpha - 2\alpha = -\frac{1}{2} - \alpha$.Since $\beta = -\frac{1}{2} - \alpha$,the other root is $4\alpha^3 - 3\alpha$.

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

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