Derivation of the quadratic equation

The simplest quadratic equation is: x2x^2

Translation in the y-direction gives: x2+Cx^2 + C

Translation in the x-direction gives: (x−B)2+C\left(x - B\right)^2 + C

Finally, scaling gives: A(x−B)2+CA\left(x - B\right)^2 + C

Solving this equation for 00 gives:

A(x−B)2+C=0(x−B)2=−CAx=B±−CA\begin{split} A\left(x - B\right)^2 + C &= 0 \\ \left(x - B\right)^2 &= \frac{-C}{A} \\ x &= B \pm \sqrt{\frac{-C}{A}} \end{split}

The “default” form of the quadratic equation is ax2+bx+cax^2 + bx +c, so let’s rewrite to this form:

A(x−B)2+C=Ax2−2ABx+AB2+CA\left(x - B\right)^2 + C = Ax^2 - 2ABx + AB^2 + C

From this, we can deduce:

a=A  ⟹  A=ab=−2AB  ⟹  B=−b2ac=AB2+C  ⟹  C=−b24a+c\begin{aligned} a &= A &\implies A &= a \\ b &= -2AB &\implies B &= \frac{-b}{2a} \\ c &= AB^2 + C &\implies C &= -\frac{b^2}{4a} + c \end{aligned}

Combine these equations to get:

x=B±−CA=−b2a±b24a2−ca=−b2a±b24a2−4ac4a2=−b±b2−4ac2a\begin{split} x &= B \pm \sqrt{\frac{-C}{A}} \\ &= \frac{-b}{2a} \pm \sqrt{\frac{b^2}{4a^2} - \frac{c}{a}} \\ &= \frac{-b}{2a} \pm \sqrt{\frac{b^2}{4a^2} - \frac{4ac}{4a^2}} \\ &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{split}