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MathGroup Archive 2007

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Re: Quadratic form: symbolic transformation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76812] Re: [mg76801] Quadratic form: symbolic transformation
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Mon, 28 May 2007 00:51:17 -0400 (EDT)
  • Reply-to: hanlonr at cox.net

q1 = r*x^2 + s*x + t;

q2 = u*(x + v)^2 + w;

Solve[CoefficientList[q1, x] == CoefficientList[q2, x], {u, v, 
    w}][[1]] // Simplify

{w -> t - s^2/(4*r), u -> r, v -> s/(2*r)}


Bob Hanlon

---- "Dr. Wolfgang Hintze" <weh at snafu.de> wrote: 
> Hello,
> 
> this is a simple question but perhaps I can get here some information
> towards a more apropriate way of using Mathematica.
> 
> I take a very simple example: I would like to write the quadratic form
> 
> q1 = R*x^2 + R*x + T
> 
> in the form
> 
> q2 = u*(x+v)^2 + w
> 
> How can I find u, v, and w from R, S, and T?
> 
> I'm sure there must be some symbolic way (using a sufficient amount of 
> _'s) to answer this question.
> 
> My (cumbersome) procedure compares coefficients and looks like this
> 
> (* writing down lhs == rhs)
> In[112]:=
> q = R*x^2 + S*x + T == u*(x + v)^2 + w
> Out[112]=
> T + S*x + R*x^2 == w + u*(v + x)^2
> 
> (* as q must be an identiy in x, i.e. must hold for all x, I compare 
> coefficients at x=0 *)
> In[113]:=
> eq1 = q /. {x -> 0}
> Out[113]=
> T == u*v^2 + w
> In[114]:=
> eq2 = D[q, x] /. {x -> 0}
> Out[114]=
> S == 2*u*v
> In[115]:=
> eq3 = D[q, {x, 2}] /. {x -> 0}
> Out[115]=
> 2*R == 2*u
> In[119]:=
> t = First[Solve[{eq1, eq2, eq3}, {u, v, w}]]
> Out[119]=
> {w -> (-S^2 + 4*R*T)/(4*R), u -> R, v -> S/(2*R)}
> 
> (* writing down the result explicitly *)
> In[120]:=
> q /. t
> Out[120]=
> T + S*x + R*x^2 == (-S^2 + 4*R*T)/(4*R) + R*(S/(2*R) + x)^2
> In[122]:=
> Simplify[q /. t]
> Out[122]=
> True
> 
> Thanks in advance for any hints.
> Regards,
> Wolfgang 
> 
> 



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