a. Consult an algebra textbook and look up the procedure of completing the square. Solve al-Khwrizms equation

a. Consult an algebra textbook and look up the procedure of completing the square. Solve al-Khwârizmî’s equation in Problem 59 by completing the square.

b. Do you think the translation of the solution in Problem 59b better describes what you did in solving the equation in Problem 59a or what you did in solving the equation in part a?

Data from Problem 59

Al-Khwârizmî solved the equation x² + 10x = 39

a. Solve this equation.

b. Here is a translation of his solution:

. . . a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: What is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5, leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

Follow these directions to find a solution. Show each step using modern algebraic notation.