A Square of Things Quadratic Equations By: Ellen Kramer

Algebra from the Beginning Year 825: Muhammad Ibn Musa Al- Khwarizmi 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= b 2 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: ax 2 + bx = c & ax 2 + c = bx Became ax 2 + bx + c = 0

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.” Compute: = = = = 3 Question: “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? Translate: –Unknown: x  “root of the square x 2 “ –“ten roots of the square”  10x Equation: x x = 39 Solutions Today Cont. Quadratic formula: X= -b + b 2 + 4c 2

Explanation of Method Using a Geometric Argument x2x2 10x x x x x x x x2x2 x2x x 25 10

Questions? Quadratic formula: X= -b + b 2 + 4ac 2 Thanks!