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# Target Six

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### Roots and Coefficients

### 8 Methods for Three by One

### An Introduction to Complex Numbers

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions

This solution is from Arun Iyer, SIA High School and Junior College.

The given equation is $x^{3/2} - 8x^{-3/2} = 7$.

Multiply throughout by $x^{3/2}$and rearranging we get

$(x^{3/2})^2 - 7x^{3/2} - 8 = 0$

This can be factorised as:

$(x^{3/2} - 8)(x^{3/2} + 1) = 0$

Case 1:

Consider,

$x^{3/2} + 1 = 0$ or $x^{3/2} = -1$

Squaring both sides, $x^3 = 1$ so the three cube roots of unity are solutions.

Now

$ 1 = e^{2k\pi i} $

(for $k=0,1,2,3$...) therefore $ x = e^{{2k\pi i} /3} $ .Putting $k=0$, $1$, $2$ we get:

$x = e^0$ (which is $1$) or

$ x = exp({\pm({2\pi i} /3)}) $

.Case 2:

Consider,

$x^{3/2}- 8 = 0$ or $x^{3/2} = 8$

Squaring both sides, $x^3 = 64$ so the three cube roots of $64$ are solutions.

Now

$ x^3 = 64 e^{2k\pi i} $

(for $k=0,1,2,3$...) therefore

$ x = 64^{1/3} e^{{2k\pi i}/3} $ .

Putting $k=0,1,2$ we get: $x = 4$ or

$ x = 4exp({\pm{2\pi i} /3}) $

.So the six roots are:

- $x_1 = 1$
- $x_2 = \exp({2\pi i} /3) = \cos (2\pi /3) + i\sin(2\pi /3) = -1/2 + i {\sqrt3}/2 $
- $x_3 = \exp({-2\pi i} /3) = \cos (-2\pi /3) + i\sin(-2\pi /3) = -1/2 - i {\sqrt3}/2 $
- $x_4 = 4$
- $x_5 = 4\exp({2\pi i} /3) = 4\cos (2\pi /3) + 4i\sin(2\pi /3) = -2 + 2i {\sqrt3} $
- $x_6 = 4\exp({-2\pi i} /3) = 4\cos (-2\pi /3) + 4i\sin(-2\pi /3) = -2 - 2i {\sqrt3} $

Substituting $x=1$ into the given equation we need to recognise that $1^{3/2}$ has two values $+1$ and $-1$ so that whereas $x^{3/2} = 1$ does not satisfy the equation the other value $x^{3/2} = -1$ does satisfy it and hence $x=1$ is a solution of the equation.

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

A short introduction to complex numbers written primarily for students aged 14 to 19.