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If the product of the roots of $4{x^3} + 16{x^2} - 9x - a = 0$ is 9, then find the value of a.

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Last updated date: 23rd May 2024
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Answer
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Hint- Use product of roots in cubic equation${\text{ = }}\dfrac{{ - {\text{ constant term}}}}{{{\text{coefficient of }}{x^3}}}$

As we know in cubic equation the product of roots is${\text{ = }}\dfrac{{ - {\text{ constant term}}}}{{{\text{coefficient of }}{x^3}}}$
Given cubic equation is $4{x^3} + 16{x^2} - 9x - a = 0$
Let the roots of this equation be $\alpha {\text{, }}\beta {\text{, }}\lambda $
And it is given that the product of roots is 9.
$
   \Rightarrow \alpha \beta \lambda = \dfrac{{ - {\text{ constant term}}}}{{{\text{coefficient of }}{x^3}}} = \dfrac{{ - \left( { - a} \right)}}{4} = 9 \\
   \Rightarrow \dfrac{a}{4} = 9 \Rightarrow a = 36 \\
$
So, the value of the constant term $a$ is 36.
So, this is the required answer.

Note- In such types of questions the key concept we have to remember is that always remember the formula of product of roots of the cubic equation, so use this property to calculate the value of product of roots in terms of $a$then equate this value to the value which is given we will get the required answer.