u1[x_] = gamma1 + x*beta1 + 1/2*alpha1*x*x;

u2[x_] = gamma2 + x*beta2 + 1/2*alpha2*x*x;

Solve[{u1[-1]==0, u2[1]==0, u1[0]==u2[0], beta1/alpha1==beta2/alpha2},
{gamma1, gamma2, beta1, beta2}]