Solve $x^{3} - 9 x^{2} + 14 x + 24 = 0$ , given that it has two roots are in the ratio of $3 : 2$ .

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Solution

$x^{3} - 9 x^{2} + 14 x + 24 = 0$

Let ratio be k, then roots are $3 k : 2 k$

Here, $x = - 1$ is a root if $- 1 = 2 k$ then other root $= - \frac{3}{2}$

And if $- 1 = 3 k$ , other root $= - \frac{2}{3}$

But roots cannot be rational here.

Now sum of root $= - \frac{b}{a}$

$= - 1 + 2 k + 3 k = 9$

$5 k = 10$
$k = 2$

Roots are $4, 6, - 1$

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