We can define a certain parabola by saying it goes through the points (0,0), (1/2,1), (1,0). I know the answer to this. However I don't understand why this method fails:

Question

empty · Answer

$$\left[ \begin{array}c 0 & 0 &  0\1/4 & 1/2 &  1\1 & 1 &  1\\end{array} \right]\left[ \begin{array}c  a\ b\ c\\end{array} \right] =\left[ \begin{array}c  0\ 1\ 0\\end{array} \right]$$ I can't invert this matrix!

anonymous · Answer

i can't make head or tails of what that is supposed to do

empty · Answer

Waittt.t.... I just realized my problem

empty · Answer

I put a full 0 vector in that top column, the last one should be a 1.

anonymous · Answer

this is way way too much work to solve this very easy question, right?

empty · Answer

Ahhh so it just represents the system of equations from plugging in the points to:

$ax^2+bx+c = f(x) $

Yeah you're right the best way I found to do this is we are given two roots so we write:
f(x) = k(x-0)(x-1) and plug in the point f(1/2)=1 to solve for k.