The general equation of a parabola with an axis parallel to Ox is (y - y0)^2 = 4a(x - x0), where the vertex is (x0, y0).
Here the vertex lies on the line y = x, so we can write it as
(y - c)^2 = 4a(x - c)
The parabola goes through the points (6 , -2) and (3 , 4). We get two equations to solve for a and c.
(4 - c)^2 = 4a( 3 - c) and ( -2 - c)^2 = 4a(6 - c)
(4 - c)^2 = 4a( 3 - c)
=> 16 + c^2 - 8c = 12a - 4ac ...(1)
( -2 - c)^2 = 4a(6 - c)
=> 4 + c^2 + 4c = 24a - 4ac ...(2)
(1) - (2)
=> 12 - 12c = -12a
=> c - 1 = a
=> c = a + 1
Substituting in (2)
=> 4 + (a + 1)^2 + 4(a + 1) = 24a - 4a(a + 1)
=> 4 + a^2 + 1 + 2a + 4a + 4 = 24a - 4a^2 - 4a
=> 5a^2 - 14a + 9 = 0
=> 5a^2 - 9a - 5a + 9 = 0
=> a(5a - 9) - 1(5a - 9) = 0
=> (a - 1)(5a - 9)
=> a = 1 and a = 9/5
c = 2 and c = 14/5
The equation of the parabola is (y - 2)^2 = 4(x - 2) and (y - 14/5)^2 = (36/5)(x - 14/5)