What is the relationship between three points on a quadratic curve and the curves coefficients? In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$?
I am asking this because I have a linear algebra problem that says:
The curve $y=ax^2+bx+c$ passes through the above points. Show that the coefficients a, b, c are a solution of the system of linear equations whose augmented matrix is 
$(x_1)^2$  $x_1$  $1$  $y_1$
$(x_2)^2$  $x_2$  $1$  $y_2$
$(x_3)^2$  $x_3$  $1$  $y_3$
So I am thinking to prove this, I would have to solve the matrix and come up with equations for a,b, and c that are already well-known formulas? Is that how I should be approaching this? If so can you tell me what the formulas should be so I can confirm?

The problem does not require you to come up with some other formula or method to find $a,b,c$. 
Instead, it asks you to give a mathematical proof that this formula finds $a,b,c$. To accomplish this:
First, start with the hypothesis, which you know to be true: assume that the curve $y = ax^2 + bx + c$ passes through the point $(x_1,y_1)$, and that it passes through the point $(x_2,y_2)$, and that it passes through the point $(x_3,y_3)$. 
Next, use the hypothesis. The statement "the curve $y=ax^2+bx+c$ passes through the point $(x_1,y_1)$" is known to be true, and you translate that statement into a mathematical equation which you may then conclude to be true.
Can you take it from here?
