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Solve the equations 6x3 – 11x2 + 6x – 1 = 0, given that the roots of each are in H.P. |
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Answer» Given equation is 6x3 – 11x2 + 6x –1 = 0 –––– (1) Put y = 1/x so that 6/y3 – 11/y2 + 6/y –1 = 0 6 – 11y + 6y2 – y3 = 0 y3 – 6y2 + 11y – 6 = 0 ––– (2) Roots of (1) are in H.P. ⇒ Roots of (2) are in A.P. Let a – d , a, a + d be the roots of (2) Sum = a – d + a + a + d = 6 3a = 6 ⇒ a = 2 Product = a(a2 – d2) = 6 2(4 – d2) = 6 4 – d2 = 3 ⇒ d2 = 1, d = 1 a – d = 2 – 1 = 1, a = 2, a + d = 2 + 1 = 3 The roots of (2) are 1, 2, 3 The roots of (1) are 1, 1/2, 1/3 |
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