8/22/2019 Quadratic Treasures from 10th Century Persia - Author Tony Foster

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Quadratic Treasures in 10th Century PersiaBy Tony Foster

Cutting edge technology makes it easy to forget that we are standing on the shoulders of

the extraordinary who came before us. The foundation of computers, cell phones, andadvances in science can be traced back to ancient mathematicians and their scrawlings ontimeworn papyrus. It was common for early practitioners of numbers to ink visuallandscapes of geometric figures to solve basic equations. Below are a series of steps tosolve a quadratic equation laid out by Euclid and further expanded upon by Omaral'Khayyamm, a poet and mathematician in 10th century Persia.Omar believed that anyalgebraic equation could be solved visually through geometry.

X2 + 7X + 12

X2

+ 7X = -12

1. Construct a square with side X. This geometrically represents X2 (the area).

2. Add a rectangle with side 7 and side X and place it next to X2. The square and therectangle aptly illustrate the equation X2 + 7X.

8/22/2019 Quadratic Treasures from 10th Century Persia - Author Tony Foster

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3. Slice the rectangle 7X in half and move one piece to the base of square X2. Notice thegeometric visual still denotes X2 + 7X.

4. Complete the square as shown in the diagram. The area of the total square is (X +3.5)2. Add 3.52 to -12 to make both sides of the equation equal. The algebraic equivalentof this visual is (X + 3.5)2 = -12 + 3.52

(X + 3.5)2 = -12 + 3.52

(X + 3.5)2 = -12 + 12.25

(X + 3.5)2 = .25 (X + 3.5)2 = .25

X + 3.5 = .25

X = -3.5 .5

X1 = -3 X2 = -4