InterviewSolution
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InterviewSolution
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Find all real value of a for which the equation `x^4 + (a - 1)x^3+ x^2 + (a - 1)x + 1 = 0` possesses at least two distinct positive roots |
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Answer» We have ltbegt `x^(4) - (a+ 1) x^(3)+ x^(2) + (a + 1) x- 2 = 0` Clearly, x = 1 and x = - 1 satisfy the above equation `because` Given equation is `(x^(2) - 1) (x^(2) - (a + 1) x + 2) = 2` So, `f(x) = x^(2) - (a + 1)x + 2 = 0` should have at least one positive real root . Since product of roots of above equation is 2, both the roots have same sign. If equation has two positive roots, then `Dge0` and sum of roots is positive (as product is already positive ). `therefore (a + 1)^(2) - 8 ge - and a + 1 gt 0` `rArr a le -1 -2 sqrt(2) - 1 and a gt - 1` `therefore a in [2sqrt(2) - 1 , infty)` . |
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