Imaginary roots (1)

Algebra Level 3

x = n 3 x = \sqrt[3]{n}

x x has 3 3 solutions: x 1 , x 2 x_{1}, x_{2} and x 3 x_{3}

x 1 = y x_{1} = y

What is the sum of x 2 x_{2} and x 3 x_{3} ?

Hint: Think of the complex plane

-y 0 1 y n

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2 solutions

Jack Rawlin
Oct 12, 2015
  1. On the complex plane the roots of a number - whether real or complex - are always evenly spaced apart, meaning all the roots sum to 0 0 . This makes it so we can express this question as an equation.

x 1 + x 2 + x 3 = 0 x_{1} + x_{2} + x_{3} = 0

  1. We know x 1 = y x_{1} = y from the question so we can substitute that in.

y + x 2 + x 3 = 0 y + x_{2} + x_{3} = 0

  1. We then re-arrange to find the sum.

x 2 + x 3 = y \boxed{x_{2} + x_{3} = -y}

Kaustubh Miglani
Oct 19, 2015

this is too simple . cube both sides use identity a^3-b^3=0 where a=x and b=cube root of n u shall get (x = \sqrt[3]{n}) or the other equation applying vieta answer is -y

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