At what values of b, c, k and l do the graphs of the functions y = kx + l and y = x ^ 2 + bx + c

At what values of b, c, k and l do the graphs of the functions y = kx + l and y = x ^ 2 + bx + c intersect at points A (6; 4) and B (4; 10)?

The solution to this problem is reduced to solving two systems of linear equations. If both graphs pass through the given points, then the coordinates of the points satisfy the equations y = kx + l and y = x ^ 2 + bx + c.
{4 = 6k + l;
{10 = 4k + l.
We solve the system by subtraction (subtract the second from the first equation):
-6 = 2k;
k = -3.
Substitute k = -3 in the first equation:
4 = 6 * (- 3) + l;
4 = -18 + l;
l = 22.
y = -3x + 22.
{4 = 6 ^ 2 + 6b + c;
{10 = 4 ^ 2 + 4b + c.
We solve the system by subtraction (subtract the second from the first equation):
-6 = 20 + 2b;
-26 = 2b;
b = -13.
Substitute b = -13 in the first equation:
4 = 36 + 6 * (- 13) + c;
4 = 36-78 + c;
4 = -42 + c;
c = 46.
x ^ 2-13x + 46 = 0.