Question

Solve the equations 6x³ – 11x² + 6x – 1 = 0, given that the roots of each are in H.P.

Answer 1

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