Shortcut for roots of quadratic equation.

Question

I know how to find the roots of a quadratic equation but the process is a bit time consuming.. Is there any way by which we can tell the roots of the quadratic equation just by looking at the equation.

Answer 1

Let $ax^2+bx+c=0$ be the quadratic. I usually start by looking if:

1) $a+b+c=0$ then the roots are $1$ and $c/a$. Example: $33x^2-41x+8=0$ has $1$ and $8/33$ as solutions.

2) $b=a+c$ then the roots are $-1$ and $-c/a$. Example: $1793x^2+2016x+223$ has $-1$ and $-223/1793$ as roots.

3) for $x^2+bx+c=0$ I test the divisors of $c$. If one of them say $\alpha$ is a root then $c/\alpha$ is a root. Example $x^2-5x+6=0$ the roots are $2$ and $3$ two divisors of $6$

If nothing of the above works, I go for the quadratic formula...

Answer 2

Suppose $a=1$ and $c$ is an integer small enough that you can instantly see its divisors. If the roots are integers, they are $r$ and $c/r$ where $r$ is a (positive or negative) divisor of $c$ such that $r + c/r = -b$. Of course the roots don't have to be integers, but this case does occur often enough.

For example, for $x^2 + 14 x + 33$, the divisors of $33$ are $1, 3, 11, 33$, and $3 + 33/3 = 3 + 11 = 14$, so the roots are $-3$ and $-11$.