How to do large number of arithmetic operations

Question

I'm having trouble with doing long arithmetic operations. Generally if the equation involves addition, subtraction, division and multiplication and it's larger than 2-3 lines of A4 paper, I can be sure that I will make a mistake. I'm still in high-school and I rarely have to deal with so large equations. But now I'm learning a new type of problems - given two quadratic equations with parameter find values of the parameter such that the two equations have common root(s), and it's seems that every time I would have to solve such problem, I would be dealing with large equation.

Answer 1

Several things you can do:
1) make your calculations more slowly, do not hurry
2) check or cross-check your calculations after every 3 steps or so
3) try not to make a mistake from step N to step N+1
4) practice by solving more problems

Answer 2

Edit: Understood you wrong Updated my answer

You can find all the roots of $f(x)$ so that $f(x)=0$ and then create a system of linear equations with $g(x)$

Let $x_1,x_2$ be the roots of $f(x)$

Now solve for $a$:

g($x_1$)=0

g($x_2$)=0

Let us use your example:

$f(x)=a*x^2-x+1$

$g(x)=2ax^2-x^2+6ax+1$

$f(x)=0$

This gives us the roots:

$x_1=\frac{1}{2a}+\sqrt{\frac{1}{4a^2}-\frac{1}{a}} $

$x_2=\frac{1}{2a}-\sqrt{\frac{1}{4a^2}-\frac{1}{a}} $

Now we plug it into $g(x)$

g($\frac{1}{2a}-\sqrt{\frac{1}{4a^2}-\frac{1}{a}}$)=0

g($\frac{1}{2a}+\sqrt{\frac{1}{4a^2}-\frac{1}{a}}$)=0

This is now a system of linear equations. The only step to do, is to solve it.