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Question 18

The sum of all the roots of the equation $$\sqrt[3]{16-x^3} = 4-x$$ is ------.


Correct Answer: 4

Solution

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Cubing both sides gives $$16-x^3 = (4-x)^3$$,

$$\implies 16-x^3 = 4^3 - x^3 - 3\cdot4\cdot x(4-x) $$

$$\implies 48-12x(4-x) = 0$$,  

 $$x^2-4x+4=0$$

Sum of the roots of the quadratic equation can be obtained from the coefficients as 

$$ \dfrac{-b}{a} = 4$$

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