a parabola passes through the points (1,1) , (2,0) and (3,1)

the equation of the parabola is y=ax^2 + bx + c

a) write down a system of equations representing this parabola.

b) solve the corresponding system and hence write down the equation of the parabola

When x = 2, y = 0. Therefore, x = 2 is a solution. I'm going to derive the Eq

with the assumption that 2 is the only
solution. If my assumption is correct,
all given points will satisfy the derived Eq.

x = 2,
x - 2 =0,
(x - 2)^2 = x^2 - 4x + 4 = 0.

CHECK:
F(2) = 2^2 - 4*2 + 4 = 0, (2 , 0).
F(1) = 1^2 - 4*1 + 4 = 1, (1 , 1).
F(3) = 3^2 - 4*3 + 4 = 1, (3 , 1).

To find the equation of the parabola passing through the given points, we can substitute the coordinates into the general equation of a parabola: y = ax^2 + bx + c.

a) To write down a system of equations representing this parabola, we need to substitute the coordinates (x, y) into the equation and form three equations using the given points:

Equation 1: When x = 1, y = 1
1 = a(1)^2 + b(1) + c --> a + b + c = 1

Equation 2: When x = 2, y = 0
0 = a(2)^2 + b(2) + c --> 4a + 2b + c = 0

Equation 3: When x = 3, y = 1
1 = a(3)^2 + b(3) + c --> 9a + 3b + c = 1

So, the system of equations representing this parabola is:
a + b + c = 1 (Equation 1)
4a + 2b + c = 0 (Equation 2)
9a + 3b + c = 1 (Equation 3)

b) To solve the system and find the values of a, b, and c, we can use various methods such as substitution or elimination. Here, let's use the method of elimination.

First, subtract Equation 1 from Equation 2:
(4a + 2b + c) - (a + b + c) = 0 - 1
3a + b = -1 --> Equation 4

By subtracting Equation 1 from Equation 3, we get:
(9a + 3b + c) - (a + b + c) = 1 - 1
8a + 2b = 0 --> Equation 5

Now, we have a system of two linear equations:
3a + b = -1 (Equation 4)
8a + 2b = 0 (Equation 5)

To solve this system, we can multiply Equation 4 by 2 and rewrite Equation 5:
6a + 2b = -2 (Equation 6)
8a + 2b = 0 (Equation 5)

By subtracting Equation 6 from Equation 5, we get:
(8a + 2b) - (6a + 2b) = 0 - (-2)
2a = 2 --> a = 1

Substituting the value of a into Equation 4, we can find b:
3(1) + b = -1
3 + b = -1
b = -4

Finally, substituting the values of a and b into Equation 1, we can find c:
1 + (-4) + c = 1
-3 + c = 1
c = 4

Therefore, the equation of the parabola passing through the points (1,1), (2,0), and (3,1) is:
y = x^2 - 4x + 4