1. Introduction. Quadratic, Cubic and Quartic Equations
This hypertext deals with the solution of polynomial equations over the field of complex numbers.
The solution of quadratic equations was already known in rudimentary form to the Babylonians (~ 400 B.C.). The method was essentially one of completing the square, but they had no notion of equation, and only dealt with positive quantities. The method was developed further by Brahmagupta (598 - 665 A.D.) to admit negative quantities, and al-Khwarizmi (800 A.D.) who gave a classification of different types of quadratics.
The first book which gave the complete solution of the quadratic equation is Liber embadorium, published in 1145 by Abraham bar Hiyya Ha-Nasi. However, this too was slightly inadequate until the the algebra of complex numbers was developed in the 18th century by Leonard Euler and others.
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The quadratic equation ax2 + bx + c = 0 has complete solution
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The solution of the general cubic equation was given for the first time in 1545 by the Italian mathematician Gerolamo Cardano in his book Ars Magna.
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The cubic equation x3 + bx + c = 0 has a solution x = ∛[-c/2 + √(c2/4 + b3/27)] + ∛[-c/2 - √(c2/4 + b3/27)] The other two solutions can be found by factoring and solving the resulting cubic. The general cubic z3 + a2z2 + a1z + a0 = 0 can be transformed into the above form by the substitution z = x - a2/3 |
The solution of the general quartic equation was also published in Cardano's Ars Magna. It is actually due to Ferrari. The quartic is first transformed into the form
x4 + px2 + qx + r = 0
by a linear substitution. This can be split into the product of quadratics if it can be written as the difference of two squares:
P2 - Q2 = (P + Q)(P - Q)
If we add and subtract x2 + u2/4 , we get
(x2 + u/2)2 - {(u - p)x2 - qx + (u2/4 - r)} = 0
giving
P = x2 - u/2 , Q2 = (u - p){x2 - q/(u - p) + (u2/4 - r)/(u - p)}
The expression for Q2 is a perfect square if u satisfies the equation
q2 = 4(u - p)(u2/4 - r)
known as the resultant cubic. Full details are given in the page Quartic Equation -- from Wolfram MathWorld.
Link: An account of the history of this topic is found at
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Quadratic-etc_equations.html