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A Square of Things
Quadratic Equations
By: Ellen Kramer
Algebra from the
Beginning
• Year 825: Muhammad Ibn Musa AlKhwarizmi wrote Arabic book titled “algebra”
• Discusses the quadratic equation with a
specific problem:
– “one square, and ten roots of the same, are equal
to thirty-nine…what must be the square which,
when increased by ten of its own roots, amounts
to thirty-nine?”
Solutions in 825
• No algebraic symbolism, thus all
problems are like recipe cards
– Solution: “you halve the number of the roots, which in the
present instance yields five. This you multiple by itself; the
product is twenty-five. Add this to thirty-nine; the sum is
sixty-four. 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; the square itself is nine.
Quadratic formula:
X=
b2
b
+c 2
2
Solutions Used Today
• Early 17th Century mathematicians came up
with algebraic symbols
– Letters from the end = unknown numbers
• Example: x, y, z
– Letters from the beginning = known numbers
• Example: a, b, c
– Thomas Harriot and Rene Descartes rearranged
equations so that they always equal 0.
• Thus: ax2 + bx = c & ax2 + c = bx
Became ax2 + bx + c = 0
Solutions Today Cont.
Question: “one square, and ten roots of the same, are equal to thirtynine…what must be the square which, when increased by ten of its own
roots, amounts to thirty-nine?
• Translate:
– Unknown: x “root of the square x2 “
– “ten roots of the square”  10x
• Equation: x2 + 10x = 39
Solution: “you halve the number of the roots, which in the present instance
yields five. This you multiple by itself; the product is twenty-five. Add this to
thirty-nine; the sum is sixty-four. 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.”
Quadratic formula:
•Compute:
52 + 39 - 5 =
X= -b + b2 + 4c
25 + 39 - 5 =
2
64 - 5 =
8-5=3
Explanation of Method Using a
Geometric Argument x
x
x
10
x2
10x
x
x2
5
5x
x
5
x
x2
5x
5
5x
25
5
5x
Questions?
Quadratic formula:
X= -b + b2 + 4ac
2
Thanks!