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