Re: Quadratic form: symbolic transformation

*To*: mathgroup at smc.vnet.net*Subject*: [mg76808] Re: [mg76801] Quadratic form: symbolic transformation*From*: Carl Woll <carlw at wolfram.com>*Date*: Mon, 28 May 2007 00:49:11 -0400 (EDT)*References*: <200705270907.FAA03614@smc.vnet.net>

Dr. Wolfgang Hintze 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 > > > SolveAlways seems like it should do the trick, but I don't know how to control which variables are solved for. Looking at the version 6 help for this function, we discover the following possibilities to accomplish what you want: eqn = R x^2 + S x + T == u (x + v)^2 + w; In[18]:= Solve[! Eliminate[! eqn, x], {u, v, w}] Out[18]= {{w->(4 R T-S^2)/(4 R),u->R,v->S/(2 R)}} In[19]:= Solve[Resolve[ForAll[{x}, eqn]], {u, v, w}] Out[19]= {{w->(4 R T-S^2)/(4 R),u->R,v->S/(2 R)}} In[20]:= Reduce[ForAll[{x}, eqn], {u, v, w}, Backsubstitution -> True] Out[20]= (S == 0 && R == 0 && u == 0 && w == T) || (u == R && R != 0 && v == S/(2 R) && w == (4 R T - S^2)/(4 R)) Carl Woll Wolfram Research

**References**:**Quadratic form: symbolic transformation***From:*"Dr. Wolfgang Hintze" <weh@snafu.de>