Find $ax^2+3x+b$ that goes through the given points using the least squares The points are $ (2,1.5),(−1,1.7),(1,1.9)$.
I did:
$A = \left [ \begin{matrix}
    1 & 2 & 4 \\
    1 & -1 & 1 \\
    1 & 1  & 1
   \end{matrix} \right ] \\
x = \left [ \begin{matrix}
    b   \\
    3   \\
    a   
   \end{matrix} \right ]\\
b = \left [ \begin{matrix}
    1.5 \\
    1.7 \\
    1.9
   \end{matrix} \right ]$
I solved for $$A^TAx = A^Tb$$
I got the Cholesky decomposition of $A^TA$

Then I solved $Ly = A^Tb$ and got $$
\left [ \begin{matrix}
    51/10 \\
    16/5 \\
    48/5
   \end{matrix} \right ]$$$
The next part is where I am having a problem. Now I have to solve $L^tx = y$. I already have some values for x, so I multiplied $L^tx$ and made it equal to y and got a system of equations


How do I solve this?
 A: $3$ is not an unknown variable, so your solution is incorrect.
We have
$$
\left\{\begin{array}{rrrrr}
2^2\cdot a &+3\cdot 2&+b&= &1.5 \\
 (-1)^2\cdot a&+3\cdot(-1)&+b&= &1.7 \\
 1^2\cdot a&+3\cdot 1&+b&= &1.9 \\
\end{array}\right.
$$
or
$$
\left\{\begin{array}{rrrrr}
4a&+&b&= &-4.5 \\
a&+&b&= &4.7 \\
a&+&b&= &-1.1 \\
\end{array}\right.
$$
or
$$
Ay=d,
$$
where
$$
A=\left(\begin{array}{ll}
4 & 1\\
1 & 1\\
1 & 1\\
\end{array}\right),\quad
d=\left(\begin{array}{r}
-4.5 \\
4.7 \\
1.9 \\
\end{array}\right),\quad
y=\left(\begin{array}{r}
a \\
b \\
\end{array}\right).
$$
This system has no solutions, thus, there is no curve of the form $ax^2+3x+b$ which goes exactly through the given points. It is only possible to find a least squares solution. This solution won't make a lot of practical sense since $a+b$ can't be equal to $4.7$ and $-1.1$ at the same time even approximately.
$$
y_{ls}=A^{+}d=(A^TA)^{-1}A^Td=\left(\begin{array}{r}
-2.1 \\
3.9 \\
\end{array}\right).
$$
Hence, the answer is $-2.1x^2+3x+3.9$.
