A Square of Things Quadratic Equations

1. A Square of ThingsQuadratic Equations By: Ellen Kramer

2. 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?” Algebra from the Beginning

3. 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

4. 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

5. Solutions Today Cont. 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 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.” • Compute: • 52 + 39 - 5 = • 25 + 39 - 5 = • 64 - 5 = • 8 - 5 = 3 Quadratic formula: X= -b + b2 + 4c 2

6. Explanation of Method Using a Geometric Argument x 5 x 10 x x2 5x x x2 10x 5 5x x 5 x x2 5x 5 5x 25

7. Questions? Quadratic formula: X= -b + b2 + 4ac 2 Thanks!