1.

Solve the equation 18x³+81x²+lambda x +60=0,one root being half the sum of the other two. Hence find the value of lambda.

Answer»

Final Answer : \lambda = 121

Steps:
1) Let the roots be \alpha,\beta and \gamma
We know that,
\alpha + \beta +\gamma = \frac {-81} {18} = \frac{-9}{2} --(1)

Then , ACCORDING to the QUESTION we have,
\alpha = \frac {\beta +\gamma } {2} \\ => \alpha = \frac{1}{2}.(\frac{-9}{2}-\alpha) \\ => \alpha = \frac{-3}{2}

2) We GOT, one root of the equation as \frac{-3}{2}

So,
18 \alpha^3 +81\alpha ^2 +\lambda x +60=0 \\ =>18 \frac{(-3)}{2}^3 +81 \frac{(-3)}{2}^2+ \lambda \frac{(-3)}{2}+ 60 =0 \\ => \lambda = 121

Therefore, REQUIRED value is 121


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