Equation in Radicals as a System of Equations

What Might This Be About?

2August 2016, Created with GeoGebra

Problem

Here's a problem Imad Zak shared at his facebook page, with a credit to Adnan Y. I. Ajez. I believe the problem is old if not well known.

Equation in Radicals as a System of Equations, problem

Solution 1

Introducing $y=\sqrt{a+x}\;$ we reduce the problem of solving an equation to that of solving a system of equations:

$y=\sqrt{a+x},\\ x=\sqrt{a+y}.$

It is hard to miss the point that the system may also result as an iteration on the function $f(x)=\sqrt{a+x}\;$ that formed a $2$-loop. Looking into the properties of the function $f(x),\;$ we may observe that all the iterations starting in the domain of the function converge when $a\gt -.25$ and none forms a $2$-loop

Equation in Radicals as a System of Equations

It thus follows that for the solution of the system, it this necessary that $y=x,\;$ implying the equation $x^2-x-a=0,\;$ with the only suitable solution being $\displaystyle x=\frac{1+\sqrt{1+4a}}{2}.$

It is also clear what makes $a=-.25\;$ exceptional: for this $a,\;$ the graph of $f(x)=\sqrt{a+x}\;$ is tangent to the diagonal $y=x,\;$ instead of crossing it:

Equation in Radicals as a System of Equations, special case

Solution 2

This solution is by Rachid Moussaoui.

Assume that the solution to the equation satisfies $x\gt\sqrt{a+x}.\;$ This would imply $a+x\gt a+\sqrt{a+x}.\;$ and, subsequently,

$x\gt\sqrt{a+x}\gt\sqrt{a+\sqrt{a+x}}=x.$

A contradiction. Similarly, assuming $x\lt\sqrt{a+x}\;$ also leads to a contradiction. It follows that necessarily $x=\sqrt{a+x},\;$ etc.

Solution 3

This solution is by Marco Antônio Manetta.

With $y=\sqrt{a+x},\;$ $x=\sqrt{a+y}\;$ and, subsequently $y^2=a+x\;$ and $x^2=a+y.\;$ Taking the difference gives $y^2-x^2=x-y,\;$ or $(x+y+1)(x-y)=0.\;$ $x\;$ and $y\;$ having to be non-negatve, $x+y+1=0\;$ is impossible. Therefore, $x=y,\;$ $x=\sqrt{a+x},\;$ and the proof ends as before.

Systems of Iterated Equations

  1. Iterations on Monotone Functions
  2. Graphing Equations Is Useful
  3. Graphing Equations Is Useful, II
  4. Graphing Equations Is Useful, III
  5. Graphing Equations Is Useful, IV
  6. Graphing Equations Is Useful, V
  7. Tangential Chaos
  8. Equation in Radicals as a System of Equations
  9. Two Conditions for a Triangle to Be Equilateral

|Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

68066467