Assume $a,b,c$ are the side lengths of a triangle. Then, if $$x^2 - ax - y + b^2 + ac = 0$$$$ax - y + bc = 0,$$ find a relationship between $a^2, b^2,$ and $c^2.$
I want to verify my solution, and I also wanted to ask if I could take this even further by finding angle measures. My solution is as follows: We can set the two given equations equal to each other, as there is only one set of solutions. We get \begin{align*} x^2 - ax - y + b^2 + ac &= ax - y + bc \\ x^2 - 2ax + b^2 - bc + ac &= 0. \end{align*} We can take the discriminant of this, which comes out to be \begin{align*} (-2a)^2 - 4(1)(b^2 - bc + ac) \\ 4a^2 - 4b^2 + 4bc - 4ac \\ 4(a^2 - b^2 + bc - ac). \end{align*} However, we are given that we only have one set of solutions, so we must have $$4(a^2 - b^2 + bc - ac) = 0,$$ which further simplifies as \begin{align*} a^2 - b^2 + bc - ac &= 0 \\ (a+b)(a-b) - c(a - b) &= 0 \\ (a - b)(a + b - c) &= 0. \end{align*} We know from the Triangle Inequality that $a + b > c,$ so we must have $a - b = 0$ in order for the solution to be true. This tells us that we must have $a = b,$ so our triangle is an isosceles triangle.
However, I was thinking of trying to identify if this was an acute, obtuse, or right triangle. Is there enough information for me to do that?