Editing Quadratic formula (section)

=== Completing the square by Śrīdhara's method ===
Instead of dividing by {{tmath|a}} to isolate {{tmath|\textstyle x^2\!}}, it can be slightly simpler to multiply by {{tmath|4a}} instead to produce {{tmath|\textstyle (2ax)^2\!}}, which allows us to complete the square without need for fractions. Then the steps of the derivation are:<ref name="Hoehn1975">{{citation |last=Hoehn |first=Larry |title=A More Elegant Method of Deriving the Quadratic Formula |journal=The Mathematics Teacher |year=1975 |volume=68 |issue=5 |pages=442–443 |jstor=27960212 |doi=10.5951/MT.68.5.0442 }}</ref>

# Multiply each side by {{tmath|4a}}.
# Add {{tmath|\textstyle b^2 - 4ac}} to both sides to complete the square.
# Take the square root of both sides.
# Isolate {{tmath|x}}.

Applying this method to a generic quadratic equation with symbolic coefficients yields the quadratic formula:

<math display=block>\begin{align}
ax^2 + bx + c &= 0 \\[3mu]
4 a^2 x^2 + 4abx + 4ac &= 0 \\[3mu]
4 a^2 x^2 + 4abx + b^2 &= b^2 - 4ac \\[3mu]
(2ax + b)^2 &= b^2 - 4ac \\[3mu]
2ax + b &= \pm \sqrt{b^2 - 4ac} \\[5mu]
x &= \dfrac{-b\pm\sqrt{b^2 - 4ac }}{2a}. \vphantom\bigg)
\end{align}</math>

This method for completing the square is ancient and was known to the 8th–9th century Indian mathematician [[Śrīdhara]].<ref>Starting from a quadratic equation of the form {{tmath|1=\textstyle ax^2 + bx = c}}, Śrīdhara's method, as quoted by [[ Bhāskara II]] (c. 1150): "Multiply both sides of the equation by a number equal to four times the [coefficient of the] square, and add to them a number equal to the square of the original [coefficient of the] unknown quantity. [Then extract the root.]". {{pb}} {{harvnb|Smith|1923|loc=[https://archive.org/details/historyofmathema0001davi/page/446 {{pgs|446}}]}}</ref> Compared with the modern standard method for completing the square, this alternate method avoids fractions until the last step and hence does not require a rearrangement after step 3 to obtain a common denominator in the right side.<ref name=Hoehn1975/>