What is the derivation of completing the square?

Question

I don't want to see the geometrical way but algebraicly like how did the idea of completing the square came from?

Answer 1

Al-Khwarizmi, also known commonly as Muhammed, derived the "completing the square" method for solving a quadratic equation.

You can use it to derive the quadratic formula

y= ax^2 + bx + c = 0 divide by a

x^2 + bx/a = - c/a take half of b/a and square it. to get b^2/4a^2. add to both sides

(x^2 + bx/a + b^2/4a^2) = -c/a + b^2/4a^2

(x+ b/2a)^2 = (b^2- 4ca)/4a^2

x +b/2a = + or - (1/2a)sqr(b^2 -4ac)

x = [-b +sqr(b^2-4ac)]/2a is the quadratic formula derived using "completing the square" method

Thank Muhammed as the Muslim mathematician was commonly called. Muhammed ibn Musa Al-Khwarizmi. He wasn't the Prophet Muhammed, or any prophet, except as to mathematics, so you don't have to add PBUH. I think. The term "Algebra" also has a Muslim Arabic origin. "al-jabr" began with Persian Muhammed, mordern day Iraq, in the Middle East, in ancient Babylon. But India also claims to have originated algebra, Bhaskara made major contributions to algebra in India. Raj on the Big Bang Theory TV show never mentioned it though. Need better script writers. Algebra is just arithmetic operations, with letters instead of numbers. China's Li Jingzhai also did major algebra. Algebra and math are a type of symbolic language. It's an international language shared across any geographical or national borders.