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:

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

Leave a Reply Cancel reply