Roots of a equation

Question

Let $t\geq 0$, $p\in(0,1/2)$, $q=1-p$ fixed and consider the equation $$t=\frac{1-x-\sqrt{(1-x)^2-4 p q}}{(1-x) \sqrt{(1-x)^2-4 p q}-(1-x)^2+4 p q}.$$ How can I get the $x=1\pm\sqrt{4 p q + t^{-2}}$?

Answer 1

Put $a = 1-x, b = \sqrt{(1-x)^2-4pq}\implies a^2-b^2= 4pq, t = \dfrac{a-b}{ab-b^2}= \dfrac{a-b}{b(a-b)}= \dfrac{1}{b}\implies b^2 = \dfrac{1}{t^2}\implies (1-x)^2-4pq = \dfrac{1}{t^2}\implies 1-x = \pm\sqrt{4pq+\dfrac{1}{t^2}}\implies x = 1\pm \sqrt{4pq+\dfrac{1}{t^2}}$ as claimed.