\(a{x^2} + b{y^2} + c = 0\)

\(\frac{a}{a}{x^2} + \frac{b}{a}x + \frac{c}{a} = 0\) ; Divide both sides by a

\({x^2} + \frac{b}{a}x = - \frac{c}{a}\)

\({x^2} + \frac{b}{a}x + {\left( {\frac{b}{{2a}}} \right)^2} = \frac{{ - c}}{a} + {\left( {\frac{b}{{2a}}} \right)^2}\) ; Complete the square

\({\left( {x + \frac{b}{{2a}}} \right)^2} = \frac{{{b^2}}}{{4{a^2}}} - \frac{c}{a}\)

\({\left( {x + \frac{b}{{2a}}} \right)^2} = \frac{{{b^2} - 4ac}}{{4{a^2}}}\)

\(x + \frac{b}{{2a}} = \frac{{\sqrt {{b^2} - 4ac} }}{{\sqrt {4{a^2}} }}\)

\(x = \frac{{ - b}}{{2a}} + \frac{{\sqrt {{b^2} - 4ac} }}{{2a}}\)

\(x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\)