To write a quadratic equation that goes through the points (0,5), (2,1), and (1,2), we can use the fact that a quadratic equation has the form:
y = ax^2 + bx + c
We can substitute the coordinates of each point into this equation to get a system of three equations in the three unknowns a, b, and c. We can then solve this system to find the values of a, b, and c.
Substituting (0,5) into the equation, we get:
5 = a(0)^2 + b(0) + c
5 = c
Substituting (2,1) into the equation, we get:
1 = a(2)^2 + b(2) + c
1 = 4a + 2b + 5
Substituting (1,2) into the equation, we get:
2 = a(1)^2 + b(1) + c
2 = a + b + 5
Now we have a system of three equations:
5 = c
1 = 4a + 2b + 5
2 = a + b + 5
Solving this system, we get:
a = -2
b = 8
c = 5
Therefore, the quadratic equation that goes through the points (0,5), (2,1), and (1,2) is:
y = -2x^2 + 8x + 5