Working out the equation of a parabola

I need to solve the unknowns in the equation of a parabola in the form of
ax^2 + bx + c
The only information I know is that it must pass through the points (45, 17.5) and (75,25)
Also, at the point (45, 17.5) it has a gradient of 1.
Is this possible to do??

y = ax^2 + bx + c
dy/dx = 2ax + b
when x = 45
1 = 90a + b
b = 1-90a ..........(1)
17.5 = 2025a+45b+c.........(2)
25 = 5625a+75b+c.....(3)
17.5 = 2025a+45(1-90a)+c
25 = 5625a+75(1-90a) + c
subtract equations
7.5 = 3600a + 30(1-90a)
7.5 = 3600a+30 - 2700a
-22.5 = 900a
a = -0.025
b = 1-(-2.25) = 3.25
c = -78.125
y = -0.025x^2+3.25x-78.125

y = ax^2 + bx + c
dy/dx = 2ax + b


When x = 45, dy/dx = 1

1 = 2a * 45 + b
1 = 90a + b
b = 1 - 90a


y = ax^2 + bx + c
y = ax^2 + (1 - 90a) * x + c
y = ax^2 + x - 90ax + c

25 = a * 75^2 + 75 - 90 * a * 75 + c
25 = 5225a - 6750a + c + 75
-50 = -1525a + c
c = 1525a - 50

17.5 = a * 45^2 + 45 - 90 * a * 45 + c
17.5 = a * 2025 + 45 - 4100a + c
-27.5 = -2100a + c
2100a - 27.5 = c

1525a - 50 = 2100a - 27.5
27.5 - 50 = 2100a - 1525a
-22.5 = 575a
-45 = 1150a
-9 = 230a
-9/230 = a

b = 1 - 90a
b = 1 - 810/230
b = (230 - 810) / 230
b = -580 / 230

2100a - 27.5 = c
2100 * (-9/230) - 27.5 = c
-18900 / 230 - 55/2 = c
-18900/230 - 6325/230 = c
-25225 / 230 = c


y = (1/230) * (-9x^2 - 580x - 25225)
y = (-1/230) * (9x^2 + 580x + 25225)

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