Firstly, did you mean a*x^2+b*c+c, or perhaps a*x^2+b*x+c? (Below, I'm assuming the latter. That makes the equations linear in a,b,c.)

Those conditions won't hold exactly for **any** d,e,f,g and x0,x1. You can do things like this in Maple, to discover such restrictions,

> P := x -> a*x^2+b*x+c:
> eq1 := P(x0)=d:
> eq2 := D(P)(x0)=e:
> eq3 := P(x1)=f:
> eq4 := D(P)(x1)=g:
> solve({eq1,eq2,eq3,eq4},{a,b,c,x0,x1});

which gives,

{a = 1/4*(e^2-g^2)/(-f+d), b = 1/2*(-x1*e^2-2*g*f+2*g*d+x1*g^2)/(-f+d),
c = 1/4*(x1^2*e^2+4*f*d+4*f*x1*g-4*d*x1*g-4*f^2-x1^2*g^2)/(-f+d),
x0 = (x1*e-2*f+2*d+x1*g)/(e+g), x1 = x1}

But that's to make the equations hold exactly. (Notice the dependence of x0 on d,e,f,g,x1. It is not "free".) The title of your post suggests that you might be looking for something else, however.

Do you want to to know a method to find a,b,c which minimizes some total of squares of residuals (least squares), for given numeric d,e,f,g and x0,x1? If so, you might consider applying LinearAlgebra:-LeastSquares something like this,

> A,B := LinearAlgebra:-GenerateMatrix({eq1,eq2,eq3,eq4},[a,b,c]);
[2 x0 1 0]
[ ] [e]
[2 x1 1 0] [ ]
[ ] [g]
A, B := [ 2 ], [ ]
[x0 x0 1] [d]
[ ] [ ]
[ 2 ] [f]
[x1 x1 1]

Notice that the above is a matrix form of the equation. Multiplying out the equivalent matrix form shows that's so.

> A . <a,b,c> - B;
[ 2 a x0 + b - e ]
[ ]
[ 2 a x1 + b - g ]
[ ]
[ 2 ]
[a x0 + b x0 + c - d]
[ ]
[ 2 ]
[a x1 + b x1 + c - f]

> sol := LinearAlgebra:-LeastSquares(eval(A,[e=0.1,g=0.2,d=0.3,f=0.4,x0=1,x1=2]),
> eval(B,[e=0.1,g=0.2,d=0.3,f=0.4,x0=1,x1=2]));
[0.0499999999999999681 ]
[ ]
sol := [-0.0099999999999998129]
[ ]
[ 0.239999999999999797 ]

I chose some specific numeric values of d,e,f,g,x0,x1 at which to evaluate A and B in doing the above LeastSquares call. The entries of **sol** are the computed values for a,b,c.

Now evaluate the equations at those same supplied data points.

> eval({eq1,eq2,eq3,eq4},[e=0.1,g=0.2,d=0.3,f=0.4,x0=1,x1=2]);
{2 a + b = 0.1, 4 a + b = 0.2, a + b + c = 0.3, 4 a + 2 b + c = 0.4}

Now evaluate the last by the least squares solution, to see how well it fits.

> eval(%,{a=sol[1],b=sol[2],c=sol[3]});
{0.09000000000 = 0.1, 0.1900000000 = 0.2, 0.2800000000 = 0.3,
0.4200000000 = 0.4}

What's being done above is this: for given numeric d,e,f,g and x0,x1 it finds the a,b,c which minimizes error in eq1,eq2,eq3,eq4 by least squares. Or at least, that was my intention. There are other commands in Maple which will do something like the steps above, and which can even quantify the calculated least squares error. But it seemed to me that you were wondering how to set it up by hand (for which it is the creation of A,B above that is key.)

acer