Question

Solve the equation x³ – 9x² + 14x + 24 = 0 if it is given that two of its roots are in the ratio 3 : 2

Answer 1

Let the roots are 3α, 2α, β

Sum of the roots are

3α + 2α + β = 9

5α + β = 9  ........(1)

Product of two roots

3α(2α) + 2α(β) + β(3α) = 14

6α² + 5αβ = 14  ........(2)

Product of three roots

(3α)(2α)β = – 24

α²β = – 4  .......(3)

(1) ⇒ β = 9 – 5α

(2) ⇒ 6α² + 5α(9 – 5α) = 14

6α² + 45α – 25α² = 14

– 19α² + 45α – 14 = 0

19α² – 45α + 14 = 0

`(alpha - 2)(alpha - 7/19)` = 0

α = 2 or α = `7/19`

If α = 2, β = 9 – 5(α)

= 9 – 5(2)

= 9 – 10

= – 1

Roots are 3α, 2α, β

3(2), 2(2), – 1

(i,e.,) 6, 4, – 1

If α = `7/19`

β = `9 - 5(7/19)`

= `(136/19)`

Roots are 3α, 2α, β (i,e.,) `21/19, 14/19, 136/19`