Editing Quadratic formula (section)

===By using algebraic identities===
The following method was used by many historical mathematicians:<ref>{{citation |doi = 10.1080/00207390802642237|title = The legacy of Leonhard Euler – a tricentennial tribute | year = 2009|last1 = Debnath|first1 = Lokenath|journal = International Journal of Mathematical Education in Science and Technology|volume = 40|issue = 3|pages = 353–388 | s2cid = 123048345}}</ref>

Let the roots of the quadratic equation {{tmath|1=\textstyle ax^2 + bx + c = 0}} be {{tmath|\alpha}} and {{tmath|\beta}}. The derivation starts from an identity for the square of a difference (valid for any two complex numbers), of which we can take the square root on both sides:

<math display=block>\begin{align}
(\alpha - \beta)^2 &= (\alpha + \beta)^2 - 4 \alpha\beta \\[3mu]
\alpha - \beta &= \pm\sqrt{(\alpha + \beta)^2 - 4 \alpha\beta} .
\end{align}</math>

Since the coefficient {{tmath|a \neq 0}}, we can divide the quadratic equation by {{tmath|a}} to obtain a [[monic polynomial|monic]] polynomial with the same roots. Namely,

<math display=block>
x^2 + \frac{b}{a}x + \frac{c}{a}
= (x - \alpha)(x - \beta)
= x^2 - (\alpha + \beta)x + \alpha\beta .
</math>

This implies that the sum {{tmath|1= \alpha + \beta = -\tfrac ba}} and the product {{tmath|1= \alpha\beta = \tfrac ca}}. Thus the identity can be rewritten:

<math display=block>
\alpha - \beta
= \pm\sqrt{\left(-\frac{b}{a}\right)^2-4\frac{c}{a}}
= \pm\frac{\sqrt{b^2 - 4ac}}{a} .
</math>

Therefore,

<math display=block>\begin{align}
\alpha &= \tfrac12(\alpha + \beta) + \tfrac12(\alpha - \beta)
        = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}, \\[10mu]
\beta  &= \tfrac12(\alpha + \beta) - \tfrac12(\alpha - \beta)
        = -\frac{b}{2a} \mp \frac{\sqrt{b^2 - 4ac}}{2a}.
\end{align}</math>

The two possibilities for each of {{tmath|\alpha}} and {{tmath|\beta}} are the same two roots in opposite order, so we can combine them into the standard quadratic equation:
<math display=block> x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .</math>