Method 1

${\displaystyle Ax^{2}+Bx+C=0}$

${\displaystyle x^{2}+{\frac {B}{A}}x+{\frac {C}{A}}=0}$

${\displaystyle x=-\alpha \pm \lambda }$

Where

${\displaystyle \alpha =-{\frac {B}{2A}}}$

${\displaystyle \beta =-{\frac {C}{A}}}$

${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

Depending on the value of ${\displaystyle \lambda }$ the equation will have the following root

One Real Root

${\displaystyle -\alpha =-{\frac {B}{2A}}}$

Two Real Roots

${\displaystyle -\alpha \pm \lambda }$

${\displaystyle -{\frac {B}{2A}}\pm {\sqrt {\frac {B^{2}-4AC}{2A}}}}$

Two Complex Roots

${\displaystyle -\alpha \pm j\lambda }$

${\displaystyle -{\frac {B}{2A}}\pm j{\sqrt {\frac {B^{2}-4AC}{2A}}}}$

Method 2

${\displaystyle ax^{2}+b=0}$

${\displaystyle x^{2}+{\frac {b}{a}}=0}$

${\displaystyle x=\pm {\sqrt {{b}{a}}}}$

${\displaystyle x=\pm j{\sqrt {\frac {b}{a}}}}$

Method 3

${\displaystyle ax^{2}-b=0}$

${\displaystyle x^{2}-{\frac {b}{a}}=0}$

${\displaystyle x=\pm {\sqrt {\frac {b}{a}}}}$

${\displaystyle x=\pm {\sqrt {\frac {b}{a}}}}$