The
general quadratic (degree 2) equation is of the form _{} and can be solved
using the quadratic formula. Egyptian, Mesopotamian, Chinese, Indian, and Greek
mathematicians solved various types of quadratic equations, as did Arab
mathematicians of the ninth through the twelfth centuries. These Arab
mathematicians lived in an empire which extended from Spain and Morocco in the West to Pakistan in the East, and which was unified by a common scientific
language, Arabic, and a common religion, Islam.

One
of the earliest of the Arab mathematicians, Muhammad ibn Musa al-Khwarizmi
(approximately 780-850 CE), was employed as a scholar at the House of Wisdom in
Baghdad in present day Iraq. Al-Khwarizmi wrote a book on the subjects of *al-jabr*
and *al-muqabala. *Al-Khwarizmi's word *al-jabr* eventually became
our word *algebra,* and, of course, the subject of his book was what we
call algebra. Incidentally, the name *al-Khwarizmi* became our word *algorithm*
(this one is a bit more of a stretch!). In his algebra book, al-Khwarizmi
solves the following problem (Katz, 245):

*What must
be the square which, when increased by ten of its own roots, amounts to 39?*

The equation al-Khwarizmi wanted to solve is _{}. Today,
we probably would solve this equation by subtracting 39 from both sides, then
factoring. Al-Khwarizmi instead used the technique of completing the square. To
complete the square, we add (10/2)^{2} = 5^{2} = 25 to both
sides of the equation, obtaining _{} The lefthand side thus
becomes a perfect square (as does the right-hand side, in this example), and we
can write the equation as _{} To solve the equation, we
take the square root of each side, obtaining _{} or _{} so that _{} or _{} Al-Khwarizmi
did not consider the negative solution; his solution was _{} Since the original
problem asked for the square, he also noted that _{}

Al-Khwarizmi
had mastered Euclid’s *Elements* (*c.* 300 BCE)
and he, like Euclid, viewed what we would call algebra very
geometrically. As a result, when al-Khwarizmi solved the equation _{} by
completing the square, he completed an actual square, as shown at right. The
solid line portion of the figure has area _{} Just as the expression _{} is not
yet a perfect square, the solid-line figure representing it also is not yet a
perfect square. To complete the square, we sketch in the dotted lines in the
lower righthand corner, adding a square region of area 5^{2} = 25 to
the figure. Notice that the completed square has side length _{} so has
area _{} On
the other hand, if we compute the area piece by piece, the completed square has
area _{} (That
_{} was
given in the original problem.) Hence, _{} Al-Khwarizmi then would
conclude that _{} so that _{} and _{}

Al-Khwarizmi gave his instructions for solving the problem in words rather than symbols, as follows (Katz, 245):

* *

*What
must be the square which, when increased by ten of its own roots, amounts to
39? The solution is this: You halve the number of roots, which in the present
instance yields five. This you multiply by itself; the product is 25. Add this
to 39; the sum is 64. Now take the root of this which is eight, and subtract
from it half the number of the roots, which is five; the remainder is three.
This is the root of the square which you sought for.*

Al-Khwarizmi lived in Baghdad, in the heart of what had been ancient Mesopotamia, and the geometric techniques he used to solve various quadratic equations, together with the verbal instructions he gave for solving them, were quite similar to those of the much earlier Mesopotamian mathematicians (Katz, 246).

Later Arab mathematicians and Renaissance Italian mathematicians were interested in solving cubic equations; they succeeded in solving these equations by---you guessed it!---"completing the cube."

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**Problems from
al-Khwarizmi’s al-Jabr:**

**1.** Translate the following problem
into a modern algebraic equation. Find both solutions of the equation using the
method of completing the square, then draw the completed square representing
the equation.

*What
must be the square which, when increased by eight of its own roots, amounts to
20?*

2. In order to solve
the following problem, first draw a diagram with the length of the side of the
square denoted by *x.* Work out your own solution, then explain how
al-Khwarizmi obtained the equation _{}.

* *

*Given
an isosceles triangle with base 12 and legs each equal to 10, inscribe a square
inside the triangle with one side along the base and the other two vertices on
the legs. What is the side of the square?*

**Instructor Notes**

**Objective: **Students will understand geometrically
the technique of completing the square.

**Solutions:**

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**1.** The equation to be solved is _{}. To
complete the square algebraically, we add (8/2)^{2} = 4^{2} =
16 to each side of the equation, obtaining _{} = _{} or _{} Then _{} and _{} or _{} Or,
students may argue geometrically as follows. The completed square shown at
right has side length _{} so has area _{} However,
if we compute the area piece by piece, the completed square has area _{} = _{} = _{} (That _{} was
given.) Hence, _{} so that _{} and _{} or _{}

**2.** Bisecting the isosceles triangle with the altitude
creates two right triangles, each with sides of lengths 6, 8, and 10 units. (Use
the Pythagorean Theorem or knowledge of 3-4-5 right triangles to see that the altitude is 8 units.)
Therefore, the area of the large triangle is 48 square units. Letting *x*
be the length of the side of the square, the square has area *x*^{2}.
The two small triangles on either side of the square each have base length _{} and
height *x,* so that their area together is _{}. The triangle above the
square has base length *x* and height _{}, so its area is _{}. The sum
of the three areas is the total area, 48 square units, so that _{}. This
equation can be simplified to 10*x* = 48, or *x* = 4.8 units.

**References: **Activity from *Lengths, Areas, and
Volumes, *by J. Beery, C. Dolezal, A. Sauk, and L. Shuey, in *Historical
Modules for the Teaching and Learning of Secondary Mathematics,* Mathematical
Association of America, Washington, D.C., 2003.

Katz, Victor J.,
*A History of Mathematics: An Introduction,* Addison-Wesley, Reading, Massachusetts, 1998.

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