Here is an interesting mathematical puzzle alike problem involving the use of Egyptian fractions, whose solution sufficiently uses the basic algebra.

## Problem

Let a, b, c, d and e be five non-zero complex numbers, and;

$a + b + c + d + e = -1$ … (i)

$a^2+b^2+c^2+d^2+e^2=15$ …(ii)

$\dfrac{1}{a} + \dfrac{1}{b} +\dfrac{1}{c} +\dfrac{1}{d} +\dfrac{1}{e}= -1$ …(iii)

$\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{1}{d^2}+\dfrac{1}{e^2}=15$ …(iv)

$abcde = -1$ …(v).

Solve and find the values of a, b, c, d and e.

## Remark

According to the problems of solution of algebraic equations, we know that, if $\alpha, \beta, \ldots , \delta, \ldots$ are the roots of an equation, then $(x-\alpha)(x-\beta) \ldots (x-\delta) \ldots =0$ .

## Solution

The algebraic equation with roots (a, b, c, d, e ) is $(x – a)(x – b)(x – c)(x – d)(x – e)=0$

After multiplying terms to each other, we get the following polynomial:
$x^5-(a + b + c + d + e) \cdot x^4+(ab+ac+ad+ae+bc+bd+be+cd+ce+de)\cdot x^3-(abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde) \cdot x^2+(abcd+abce+abde+acde+bcde) \cdot x -abcde=0$
… (vi)

Squaring both sides of equation (i), we get

$a^2+b^2+c^2+d^2+e^2+ 2ab + 2ac + 2ad + 2ae + 2bc + 2bd + 2be + 2cd + 2ce + 2de = 1$
Or,
$a^2+b^2+c^2+d^2+e^2+ 2(ab + ac + ad + ae + bc + bd + be + cd + ce + de) = 1$ …(vii)

Subtracting equation (ii) from the equation (vii)

$2(ab + ac + ad + ae + bc + bd + be + cd + ce + de) = -14$
Or, $ab+ac+ad+ae+bc+bd+be+cd+ce+de=-7$ …(viii)

Now, multiplying equation (iii) by (v) :: multiplying left side by abcde and right side by -1:: we have

$bcde + acde + abde + abce + abcd =1$
or, $abcd+abce+abde+acde+bcde=1$ … (ix)

Again, multiplying equation (iv) by the square of equation (v), we get

$(bcde)^2 + (acde)^2+ (abde)^2+ (abce)^2+ (abcd)^2= 15$

Or, $(abcd)^2+ (bcde)^2+ (cdea)^2+ (deab)^2+ (eabc)^2= 15$ … (x)

Squaring $abcd + bcde +cdea +deab + eabc = 1$ we get

$(abcd)^2+(bcde)^2+(cdea^2)+(deab)^2+(eabc)^2+2abcde \cdot (abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde)=1$ … (xi)

Substitute known values in (xi):

$15 – 2(abc + abd + abe + acd + ace + ade + bcd + bce + bde + cde) = 1$

Or, $abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde =7$ … (xii)

Putting values from equation, (i), (viii), (xii), (ix) and (v) respectively, to the equation (vi), we obtain the algebraic equation
$x^5+x^4-7x^3-7x^2+x+1=0$ … (xiii).

Now, Equation (xiii) has the solutions a, b, c, d and e.

Factorizing (xiii) we get:
$(x + 1)(x^4- 7x^2 + 1) = 0$
$x = -1$ or $x^4- 7x^2+ 1 = 0$
As, $x^4- 7x^2+ 1 = 0$ has roots:

1. $\dfrac{1}{2} (3+\sqrt{5})$
2. $\dfrac{1}{2} (3-\sqrt{5})$
3. $\dfrac{1}{2} (-3+\sqrt{5})$
4. $\dfrac{1}{2} (-3-\sqrt{5})$

Thus, a, b, c, d and e are

• $-1$
• $\dfrac{1}{2} (3+\sqrt{5})$
• $\dfrac{1}{2} (3-\sqrt{5})$
• $\dfrac{1}{2} (-3+\sqrt{5})$
• $\dfrac{1}{2} (-3-\sqrt{5})$

irrespective of any distinct order due to their symmetry.

## Solving Integral Equations – (1) Definitions and Types

If you have finished your course in Calculus and Differential Equations, you should head to your next milestone: the Integral Equations. This marathon series (planned to be of 6 or 8 parts) is dedicated to interactive learning of integral equations for the beginners —starting with just definitions and demos —and the pros— taking it to the heights of problem solving.…

## Hopalong Orbits Visualizer: Stunning WebGL Experiment

Just discovered Barry Martin’s Hopalong Orbits Visualizer — an excellent abstract visualization, which is rendered in 3D using Hopalong Attractor algorithm, WebGL and Mrdoob’s three.js project. Hop to the source website using your desktop browser (with WebGl and Javascript support) and enjoy the magic. PS: Hopalong Attractor Algorithm Hopalong Attractor predicts the locus of points in 2D using this algorithm…

## What is Real Analysis?

Real analysis is the branch of Mathematics in which we study the development on the set of real numbers. We reach on real numbers through a series of successive extensions and generalizations starting from the natural numbers. In fact, starting from the set of natural numbers we pass on successively to the set of integers, the set of rational numbers…

## On World Math Day – Do A Simple Self-Test to Identify a Creative Mathematician in You

There are many mathematicians, chemists, musicians, painters & biologists among us. But they are unaware of their qualities. Only some small suggestive strokes are needed for them. Unless one tells them, they will not know how great they are. Here is a small self test. Do you enjoy music, drums and dance? Do you enjoy looking at the flowers, carpets,…