Stuck on a basic calc problem The curve $y=ax^2+bx+c$ passes through the points $A(-2,0)$ and $B(3,20)$. The slope of the normal at $A$ is $1$ and the gradient at $B$ is $9$. Find the values of $a,b,$ and $c$. 
for $A$: $0=4a+-2b+c$ 
for $A'$: $-4a+b=-1$
for $B$: $20=9a+3b+c$
for $B'$: $9=6a+b$ 
This can be solved by using a matrix; how to solve it without a matrix? 
Some hints? 
 A: Certainly you do not need matrix notation to solve your system of equations. Look at the two equations where only $a$ and $b$ occur.  The first of these equations says that $b=4a-1$. Substitute for $b$ in the second equation, and solve for $a$.  We get $a=1$. Now that you know $a$, you also know $b$.
Finally, use one of the two equations that involve $c$, and the known values of $a$ and $b$, to solve for $c$. Then check whether the $a$, $b$, and $c$ that you have found satisfy the fourth equation. They very might not: you have four conditions (and linear equations) and only three unknowns.  
Remark: The person who made up the problem carefully chose numbers such that the solution to the first three equations we considered is also a solution of the fourth equation.  If you try to make up a similar problem by choosing "random" values for the slope of the normal at $A$ and the gradient at $B$, you will most of the time end up with a system of equations that has no solution. 
The problem could have been made a little harder. For example, instead of being told the gradient at $B$, you could have been asked what that gradient is. 
A: Augmented matrix is just a simpler notation for method of elimination. You can always use method of elimination and then backward substitution to solve it.
