If $2$ roots of an equation $x^{3} - 9 x^{2} + 14 x + 24 = 0$ are in the ratio $3 : 2$ . Find the roots.

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Solution

$x^{3} - 9 x^{2} + 14 x + 24 = 0$
$∵ x = - 1$ is $a$ factor,
$x^{2} - 10 x + 24 x + 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{3} - {9 x}^{2} + 14 x + 24 - x^{3} + x^{2}_______________{- 10 x}^{2} + 14 x + 24 {- 10 x}^{2} - 10 x24 x + 24 24 x + 240$

$- 9 x^{2} + 14 x + 24 = (x + 1) (x^{2} - 10 x + 24)$
given that the roots are in ratio $3 : 2$
let $3 K$ & $2 K$ be two roots, then.
$(x - 3 k) (x - 2 k) = x^{2} - 51 x + 6 k$
comparing with $x^{2} - 10 x + 24$
$- 5 k = 10 \Rightarrow k = 2$
or $6 k^{2} = 24 \Rightarrow k^{2} = 4 = k = \pm 2$
but, we have $k = 2,$
$∴$ Roots are $6, 4$
hence, roots are $1, 6, 4.$

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