When the graph touches x axis at x = x0, we would get the roots of the equation to be equal.
Do you understand what these words mean? For example, (a) says that the graph 'touches the x- axis at 4". First that tells us that when x= 4, y= 0. But the graph "touches" the x-axis, it does NOT cross it! That means that the vertex is at (4, 0). It means that we can write the equation as y= a(x- 4)^2. Since the graph "passes through (2, 12) we have 12= a(2- 4)^2= a(-2)^2= 4a so a= 3. The equation is y= 3(x- 4).^2 We can also write that as y= 3(x^2- 8x+ 16)= 2x^2- 24X+ 48.