Actually we were told by our teacher that the quadratic equation that satisfies the conditions mentioned below has integral roots...
Consider the equation- $ax^2+bx+c=0$ in its lowest form.
- $a,b,c$ are integers
- $a = 1$ or $-1$
- $b^2-4ac$ is a perfect square
But to my surprise while solving some high school level questions i came across the equation $2x²-5x-3=0$ which has two root, $3$ and $-1/2$ out of which the first root is integral for sure..which should not be so looking at the conditions.
Now my question is: Where did my teacher or I went wrong?
If this is not the correct way to predict integral root for a quadratic equation then what is it?
The statement your teacher gave you says: "If conditions hold, then you have two integer roots".
Now if you reverse that statement, it would read like "If you don't have two integers roots (so either one, or none) then at least one of the conditions doesn't hold.
So in your example even though you found one integer root, the other wasn't, which means that at least one of the three conditions was not satisfied, in particular you had $a=2$ instead of $a=±1$ as required.
In conclusion, the statement you have tells you when you have exactly two integers roots, rather than integer roots (i.e. one integer root) in general