# 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?

• A formula is not the right approach for this problem. Note that we have $ax_i^2+bx_i+c=y_i$ for $i=1,2,3$. That directly translates to the system of linear equations, augmented matrix formulation. Sep 8, 2014 at 12:01
• Yes I understand, but I just want to know the formula so that after obtaining solutions ( in terms of x1, y1... Etc )I can compare that to the formula and that would be my proof of the fact that the coefficients of the quadratic equation are what satisfies that set of linear equations. Sep 8, 2014 at 12:08
• Unless that is not how I should be proving it, in which case, how should I do it? Sep 8, 2014 at 12:08
• But I realize that is kind of circular reasoning. Sep 8, 2014 at 12:14
• What André is saying is that the problem, as stated, only asks you to show that $(a, b, c)$ comprise a solution to a system of equations, which is not the same as asking for explicit formulas for $(a, b, c)$. The coefficient matrix, up to a reversal of column order, is called a Vandermonde matrix; explicit forms for its inverse (which always exists provided the $x_a$ are pairwise distinct) are available but are already a little messy in the $3 \times 3$ case, a hint that you should avoid solving explicitly if you can. Sep 8, 2014 at 12:15

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?

• So I should solve the matrix to get a,b and c and then substitute those into the quadratic equations for each set of coordinates? And that's the proof? Sep 8, 2014 at 12:50
• Or is the fact that the system is consistent in itself proof of the coefficients? Sep 8, 2014 at 12:56
• You still seem overly concerned with solving the system of equations for $a,b,c$. That is not your job. Your job is to verify that the system of equations is true. Sep 8, 2014 at 13:06
• No,I am sorry. I really don't know how to approach this. Sep 8, 2014 at 13:12
• Would this suffice as proof? Since the given points lie on the curve, the solution to the matrix must be a,b c to satisfy the quadratic equation. Sep 8, 2014 at 13:39