From the previous O.D.S. stories (#3, #4) on Quintic equations (degree 5) by Galois and Abel in the 19th century, we now trace back to the first breakthrough in the 16th century of the Cubic (degree 3) & Quartic (degree 4) equations with **radical** solution, i.e. expressed by 4 operations (+ – × /) and *radical***roots ** {$latex sqrt{x} , : sqrt [n]{x} $ }.

**Example:** Since Babylonian time, and in 220 AD China’s Three Kingdoms Period by 趙爽 Zhao Shuang of the state of **Wu 吳**, we knew the radical solution of Quadratic equations of degree 2 :

$latex ax^2 + bx + c = 0 $

can be expressed in radical form with the coefficients a, b, c:

$latex boxed{x= frac{-b pm sqrt{b^{2}-4ac} }{2a}}$

Are there radical solutions for Cubic equation (degree 3) and Quartic equations (degree 4) ? We had to wait till the European Renaissance…

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