How Can Points on a Parabola Determine Its Equation Coefficients?

[icesalmon]

Homework Statement

The curve y = ax² + bx + c passes through the points Q(x₁,y₁) R(x₂,y₂), S(x₃,y₃). Show that the coefficients a, b, and c are a solution of the system of linear equations whose augmented matrix is x₁² + x₁ + 1 = y₁
x₂² + x₂ + 1 = y₂ x₃² + x₃ + 1 = y₃

Homework Equations

The Attempt at a Solution

I don't know where to start, I tried creating linear equations using the points Q, R, and S and the point slope formula. but it got messy. This is from Anton's 10th edition of Linear Algebra with applications pg. 10

 

[dikmikkel]

To check if something is a solutions means:
substitute the solution in the system and check that the LHS equals the RHS.

 

[LCKurtz]

Homework Statement

The curve y = ax² + bx + c passes through the points Q(x₁,y₁) R(x₂,y₂), S(x₃,y₃). Show that the coefficients a, b, and c are a solution of the system of linear equations whose augmented matrix is x₁² + x₁ + 1 = y₁
x₂² + x₂ + 1 = y₂ x₃² + x₃ + 1 = y₃

What you have written isn't a matrix, so that isn't the form you are looking for.

Homework Equations

The Attempt at a Solution

I don't know where to start, I tried creating linear equations using the points Q, R, and S and the point slope formula. but it got messy. This is from Anton's 10th edition of Linear Algebra with applications pg. 10

You don't need the point slope form. Just write the three equations given by requiring the points $(x_1,y_1),\,(x_2,y_2),\, (x_3,y_3)$ satisfy the equation $ax^2+bx+c = y$ and think about these three equations in the three unknowns $a,b,c$. What do you get for their augmented matrix?

P.S. That system in $a,b,c$ will not be a non-linear system.