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 radicalroots {$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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Mathtuition88 