the quadratic formula

    \[ax^2+bx+c=0\]

    \[\to x^2+\frac{b}{a}x+\frac{c}{a}=0\]

Completing the square, \displaystyle (x+\frac{b}{2a})^2-\frac{b^2}{4a^2}+\frac{c}{a}=0

Substituting in y=x+\frac{b}{2a},

    \[y^2=\frac{b^2-4ac}{4a^2}\]

This is an equation with only y^2 and not y, so take the square root of both sides:

    \[y=\frac{\pm\sqrt{b^2-4ac}}{2a} \to x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

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