solutions to quadratic equation with sum = 2(k+l) and product = 3(k^2 - l^2) + kl

Question

What are the solutions to the quadratic equation with:

$ x_1 + x_2 = 2(k+l) $

$ x_1 * x_2 = 3(k^2 - l^2) + kl $

Answer 1

Note that $x_1 + x_2 =2(k+l)$ and $x_1 x_2=3(k^2-l^2)+kl$ imply that $$(x_1-x_2)^2 = (x_1+x_2)^2-4x_1x_2 = 16l^2+4kl-8k^2.$$ Consequently, $$x_1-x_2 = \pm2\sqrt{4l^2-2k^2+kl}.$$ Using $x_1+x_2=2(k+l)$, we can easily show that $$(x_1,x_2)=(k+l+\sqrt{4l^2-2k^2+kl},k+l-\sqrt{4l^2-2k^2+kl}),$$ or $$(x_1,x_2)=(k+l-\sqrt{4l^2-2k^2+kl},k+l+\sqrt{4l^2-2k^2+kl}).$$