<s>[INST] Proving Quadratic Formula purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become $\dfrac{b^2}{4a^2}$?
 [/INST] Remember how to complete the square:
$$Ax^2+Bx=A\left(x+\frac{B}{2A}\right)^2-\frac{B^2}{4A^2}$$
So now
$$ax^2+bx+c=0 ---- \text{complete square}$$
$$a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a}=-c$$
$$\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$$
$$x_{1,2}+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$$
$$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
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