Euclid II.6
(Geometric Algebra)
If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line. 
In the illustration below, the given straight line (the one getting bisected) is represented by AB, with C being its midpoint. The added line is BD. The "rectangle contained by the whole with the added straight line and the added straight line" is the product AD·BD. The "square on the half" is CB·CB. The proposition asserts that the sum
Introduce a = AB, x = BD. Then
(1) 

CB = a/2. Adding CB·CB = a^{2}/4 to (1) completes the square on the right:
a^{2}/4 + ax + x^{2} = (a/2 + x)^{2}. 
If we assume that the area of the rectangle AD·BD is given and equals b^{2}, then we obtain
(a/2 + x)^{2} = b^{2} + a^{2}/4, 
wherefrom x can be found:
(2)  x = √b² + a²/4  a/2. 
which appears to be a solution to the quadratic equation
ax + x^{2} = b^{2}. 
The square root in (2) can be found with the help of the Pythagorean theorem.
References
 T. L. Heath, Euclid: The Thirteen Books of The Elements, v. 1, Dover, 1956
Complex Numbers
 Algebraic Structure of Complex Numbers
 Division of Complex Numbers
 Useful Identities Among Complex Numbers
 Useful Inequalities Among Complex Numbers
 Trigonometric Form of Complex Numbers
 Real and Complex Products of Complex Numbers
 Complex Numbers and Geometry
 Remarks on the History of Complex Numbers
 Complex Numbers: A Dynamic Tool
 Cartesian Coordinate System
 Fundamental Theorem of Algebra
 Complex Number To a Complex Power May Be Real
 One Can't Compare Two Complex Numbers
Euclid's Elements Reference Page
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