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Quadratic Equation, A Geometric Solution
Taken from:
http://occawlonline.pearsoned.com/bookbind/pubbooks/angel_awl/chapter1/medialib/internet_projects/chap
ter6/algebra/algebra.html
but it’s easier to find it in a Google search: “Chapter 6” “Quadratics in History”.
© 2001 by Addison Wesley Longman, A division of Pearson Education
Consider this problem: A square and 20 roots is 300.
Here, a root is x and the square is x2. So this says x2 + 20x = 300
On page 2 is a geometric solution, developed by the Middle Eastern mathematician al-Khwarizmi (or Abu
Ja'far Muhammad ibn Musa Al-Khwarizmi), believed to have lived from 780 AD to 850 AD. You’ll notice
that the solution, creative in its design, requires the completion of the square.
You’ll also notice that it yields only one solution, x = 10, yet our algebraic methods will give us two
solutions:
x2 + 20x = 300
x2 + 20x – 300 = 0
(x – 10)(x + 30) = 0
x = 10, x = - 30
Since the geometric method considers only positive solutions, it makes sense that x = - 30 is not a viable
solution.
Furthermore, we could get two solutions only if the constant (on the right side of the equation) were
negative. It is unlikely, however, that negative constants would be considered at all.
I became curious, though, and asked, “What if the problem read: A square less 20 roots is 300?” How
might we approach it then?
Algebraically it would look like this:
x2 – 20x = 300
x2 – 20x – 300 = 0
(x + 10)(x – 30) = 0
x = - 10, x = 30
This time, the solution is 30. Geometrically, the solution is shown on page 3.
