f1(x):=a1*x^2+b1*x+c1;
f2(x):=a2*x^2+b2*x+c2;

r:(b1-b2)/(2*(a2-a1));fr:f1(r);

algsys([a2*x(P)^2+b2*x(P)+c2=y(P), a2*x(Q)^2+b2*x(Q)+c2=y(Q), a2*r^2+b2*r+c2=fr], [a2,b2,c2]);

(x(P)^2*y(Q)^2+sv*((2*x(P)*su*x(Q)+2*x(P)^2*su)*y(Q)-2*a1*x(P)*su*x(Q)^3+su*(2*y(P)-4*a1*x(P)^2-4*b1*x(P))*x(Q)^2+su*(2*x(P)*y(P)-2*a1*x(P)^3-4*b1*x(P)^2-8*c1*x(P))*x(Q))+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2-6*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(4*a1*c1+6*b1^2)*x(P)^2+8*b1*c1*x(P))*x(Q)^2+((-b1*x(P)^2-6*c1*x(P))*y(P)+a1*b1*x(P)^4+(6*a1*c1+b1^2)*x(P)^3+8*b1*c1*x(P)^2+8*c1^2*x(P))*x(Q))

factor((x(P)^2*y(Q)^2+sv*((2*x(P)*su*x(Q)+2*x(P)^2*su)*y(Q)-2*a1*x(P)*su*x(Q)^3+su*(2*y(P)-4*a1*x(P)^2-4*b1*x(P))*x(Q)^2+su*(2*x(P)*y(P)-2*a1*x(P)^3-4*b1*x(P)^2-8*c1*x(P))*x(Q))+((y(P)-3*a1*x(P)^2-b1*x(P))*x(Q)^2+(4*x(P)*y(P)-4*a1*x(P)^3-6*b1*x(P)^2-6*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2)*y(Q)+(-a1*y(P)+2*a1^2*x(P)^2+a1*b1*x(P))*x(Q)^4+((-4*a1*x(P)-b1)*y(P)+4*a1^2*x(P)^3+7*a1*b1*x(P)^2+(6*a1*c1+b1^2)*x(P))*x(Q)^3+(y(P)^2+(-3*a1*x(P)^2-6*b1*x(P)-2*c1)*y(P)+2*a1^2*x(P)^4+7*a1*b1*x(P)^3+(4*a1*c1+6*b1^2)*x(P)^2+8*b1*c1*x(P))*x(Q)^2+((-b1*x(P)^2-6*c1*x(P))*y(P)+a1*b1*x(P)^4+(6*a1*c1+b1^2)*x(P)^3+8*b1*c1*x(P)^2+8*c1^2*x(P))*x(Q)));

ratsimp(%);

((2*su*x(Q)^2-4*x(P)*su*x(Q)+2*x(P)^2*su)*sv+(x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)-2*c1)*x(Q)^2+(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+4*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2)

ratsimp(((2*su*x(Q)^2-4*x(P)*su*x(Q)+2*x(P)^2*su)*sv+(x(Q)^2-2*x(P)*x(Q)+x(P)^2)*y(Q)-a1*x(Q)^4+(2*a1*x(P)-b1)*x(Q)^3+(y(P)-2*a1*x(P)^2+b1*x(P)-2*c1)*x(Q)^2+(-2*x(P)*y(P)+2*a1*x(P)^3+b1*x(P)^2+4*c1*x(P))*x(Q)+x(P)^2*y(P)-a1*x(P)^4-b1*x(P)^3-2*c1*x(P)^2));