Okay, so I have a problem, which asks me to prove a condition ; i.e $(p^2=4q)$ based on the fact that the following quadratic equation has equal roots :
$$(1-q+\frac{p^2}{2})x^2+p(1+q)x+q(q-1)+\frac{p^2}{2}=0.$$
My Try Knowing the fact that a quadratic equation has equal roots if it's Discriminant is equal to $0$, I applied the same condition, but not only did my calculations become super complex, they also did not render the following proof.
My question is, why am I not getting the answer from this condition? Anyways, there has to be a shorter method to this to solve these kinds of problems, please let me know how'd you solve them.
We have $$p^2(1+q)^2-4\left(1-q+\frac{p^2}{2}\right)\left(q^2-q+\frac{p^2}{2}\right)=0$$ or $$(p^2-4q)(p^2+q^2-2q+1)=0,$$ which gives $p^2=4q$ because $p^2+(q-1)^2=0$ gives $p=0$ and $q=1$
and from here $1-q+\frac{p^2}{2}=0$, which is impossible.
Done!