Value of an algebric expression in a quadratic equation

I came across such a problem:

Given the equation \begin{equation} x^2 + \sqrt{m} x + n = 0 .\tag{1} \end{equation} If it has two equal real roots, what is the value of $(m+1)(m-1) - 2(2n - 1)$?

This is what I have done: Since the quadratic equation has two equal roots, we have \begin{equation} m - 4n = 0 ,\tag{2} \end{equation} or \begin{equation} m = 4n .\tag{3} \end{equation} However, \begin{equation} (m+1)(m-1) - 2(2n - 1) = m^2 - 4n + 1 ,\tag{4} \end{equation} I tried many ways to manipulate eq. (4), but couldn't figure out its value. I don't know how eqs. (2) or (3) can help. Is there any way out?

Solution 1:

We have

\begin{align*} (m+1)(m-1)-2(2n-1)&=m^2-1-4n+2\\ &=m^2-4n+1\\ &=m^2-m+1 \end{align*}

There's no way to further "solve" this problem. There are infinitely many values for $m$ that satisfy the given conditions.

It should be noted that if we are working with a polynomial over the reals, then we must also add the condition that $m\geq 0$, since $\sqrt\cdot$ cannot input negative values.