Question

Solve for $x:4x^2-4a^{2}x+(a^2-b^2)=0$

Answer 1

$4x^2-4a^{2}x+a^2=b^2$

$\Rightarrow {\left(2x\right)}^2-2\times 2x\times a^2+a^4-a^4+a^2=b^2$

$\Rightarrow {(2x-a^2)}^2=a^4-a^2+b^2$

$\Rightarrow 2x-a^2=\pm \sqrt{a^4-a^2+b^2}$

$\Rightarrow 2x=a^2\pm \sqrt{a^4-a^2+b^2}$

$\therefore x=\frac{a^2\pm \sqrt{a^4-a^2+b^2}}{2}$