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Let $(E_l) : x^4-4x^3+x^2(5-l^2)+4 x l^2-4l^2 = 0$ be an equation with $l$ as a parameter and $x$ as an unknown and $S_l$ the set containing all the real solutions of $(E_l)$.

Then there is a unique value of $l$ verifying $card(S_l) = 1$.

Is there a way in Mathematica to find an approximate value for this $l$ ?

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2 Answers 2

up vote 7 down vote accepted

Complex roots come in pairs therefore the equation will have a single real solution only if it has a real double root and two complex root or if it has a quadruple root.

So we require

poly = -4 l^2 + 4 l^2 x + (5 - l^2) x^2 - 4 x^3 + x^4

Solve[{poly == 0, D[poly, x] == 0}, {x, l}, Reals]

This gives us three solutions for l:

{0, -((2 + 2^(1/3))^(3/2)/Sqrt[2]), (2 + 2^(1/3))^(3/2)/Sqrt[2]}

Substituting them back into poly and finding all roots shows that only l == 0 will give double real root and two complex roots.

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I'm quite disappointed I didn't think of using the derivative. Very clever solution, thanks ! –  Skydreamer Oct 4 '13 at 14:49
With[{poly = x^4 - 4 x^3 + (5 - l^2) x^2 + 4 l^2 x - 4 l^2},
 With[{sols = 
    DeleteDuplicates@Solve[Discriminant[poly, x] == 0, l, Reals]},
  Select[
   sols,
   Length[DeleteDuplicates@Solve[(poly /. First@#) == 0, x, Reals]] ==
      1 &
   ]
  ]]

(* {{l -> 0}} *)
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@Michel E2 If I want to the equation $x^4+2(m+1)x^2 - m - 4 =0$ has three reals solutions. How can I get it? –  minthao_2011 Oct 5 '13 at 3:52
    
@minthao_2011 Should this be a separate question, not a comment? In any case, three (distinct) real solutions implies two simple and one double root. From Discriminant[x^4 + 2 (m + 1) x^2 - 4, x], one sees that only m -> -4 will yield a double root; luckily it also yields two other simple real roots. –  Michael E2 Oct 5 '13 at 12:24

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