Question

Solve the equation
6x⁴ – 35x³ + 62x² – 35x + 6 = 0

Answer 1

This is Type I even degree reciprocal equation.

Hence it can be rewritten as

`6(x^2 + 1/x^2) - 35(x + 1/x) + 62` = 0 ......(1)

y = `x + 1/x`

y² = `x^2 + 1/x^2 + 2`

y² – 2 = x^2 + 1/x^2`

(1) ⇒ 6(y² – 2) – 35y + 62 = 0

6y² – 12 – 35y + 62 = 0

6y² – 35y + 50 = 0

`(y - 5/2)(y - 10/3)` = 0

y = `5/2` or y = `10/3`

`x + 1/x = 5/2` or `x + 1/x = 10/3`

`(x^2 + 1)/x = 5/2` or `(x^2 + 1)/x = 10/3`

2x² + 2 – 5x = 0 or 3x² + 3 = 10x

2x² – 5x + 2 = 0 or 3x² – 10x + 3 = 0

x = `2, 1/2` or x = `3, 1/3`

Roots are `2, 1/2, 3` and `1/3`