Quadratic form: symbolic transformation

• To: mathgroup at smc.vnet.net
• Subject: [mg76801] Quadratic form: symbolic transformation
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Sun, 27 May 2007 05:07:38 -0400 (EDT)
• Reply-to: "Dr. Wolfgang Hintze" <weh at snafu.de>

```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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