Prove 3 distinct points lying on the some curve

[zohapmkoftid]

Homework Statement

Given points (p₁, q₁), (p_2, q₂), (p₃, q₃) in the plane with p₁, p₂, p₃ distinct, show that they lie on some curve with equation y = a + bx + cx².

It should be related to matrix but I have no idea about this question. Could anyone help? Thanks!

Homework Equations

The Attempt at a Solution

 

[hunt_mat]

One way to examine this question is given these three point you could find out, what a,b, and c is satisfied for this equation. So you know that:
$$\begin{array}{ccc} q_{1} & = & ap_{1}+bp_{1}+cp_{1} \\ q_{2} & = & ap_{2}+bp_{2}+cp_{2} \\ q_{3} & = & ap_{3}+bp_{3}+cp_{3} \end{array}$$
This is a system of linear equations with unknowns a,b and c, the system can be solved via matrix methods.

 

[zohapmkoftid]

Thanks for your help!
But I don't understand how the equation y = a + bx +cx² can be transformed into the form y = ax + bx + cx

 

[hunt_mat]

Because it can't. I made a typo, sorry! The equations should read:
$$\begin{array}{ccc} q_{1} & = & ap_{1}+bp_{1}+cp_{1}^{2} \\ q_{2} & = & ap_{2}+bp_{2}+cp_{2}^{2} \\ q_{3} & = & ap_{3}+bp_{3}+cp_{3}^{2} \end{array}$$

Mat