Quadratic Equations

1. Quadratic Equations Andrew Nystrom Math 250

2. Algebra • Where did the word come from? • A book from the year 825 • Author is Muhammad IbnMūsa Al-Khwārizmī

3. Muhammad IbnMūsa Al-Khwārizmī • From modern-day Uzbekistan • Lived in Baghdad • Studied at “The House of Wisdom” • Wrote many books; “Algebra” most famous

4. The First Quadratic Equation • “One square, and ten roots of the same, are equal to thirty-nine dirhems. That is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?” • In other words,

5. His solution • Halve the number of roots • Multiply it by itself • Add it to the LHS • Take the root of the resulting sum • Subtract half the number of roots • Answer! Essentially, we get

6. Our solution • Write it in the following form: • Then use the quadratic equation: • Then

7. Difference • Using our method, we get • Our method yields both a positive and negative root. Why? • Negative numbers were not recognized during Al-Khwārizmī’s time. • But are his and our formulas really different?

8. Difference • Basically, his method for solving quadratic equations is • With a little algebra, we can reconcile it to something like our modern version.

13. Further work • In the 16th century, Thomas Harriot and René Descartes wrote equations as • Leading to

14. Thank you !