Question

104) Use Polya's manipulatives / drawing strategy with algebra tiles to demonstrate the procedure of solving the following problems visually as done in class. (a) Factorize P(x) = 8x2 - 2x - 15 (b) Also known as the father of algebra, Muhammad ibn Musa al-Khwarizmi (780-850 CE) is the first mathematician to write an algebra book. In his treatise “Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala," al-Khwarizmi considers the following problem: What must be the square which, when increased by ten of its own roots, amounts to 39? (Katz, 245). Al-Khwarizmi then offers the following solution which is based on the Completing the Square technique: You halve the number of roots, which in the present instance yields five. This you multiply by itself; the product is 25. Add this to 39; the sum is 64. 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 (Katz, 245). *Question: Use the Completing the Square technique with algebra tiles (using the 4-step procedure as done in class) to solve the following problem posed by al-Khwarizmi: What must be the square which, when increased by eight of its own roots, amounts to 20?

Answer 1

(a) Factorize : P(x)=8n²-zu-15 Completing the square method :- stepl. Make the coefficient # 1 leading coefficient N 15 s(- 4 Step2. From Now, on we ignore the factor outside the brackets and continue working on inside the quadratic polynomial brackets. Take the coefficient of the linear tem divide it by 2 & then square it. This produces the quantity 64 Add and immedi subtract Step3 back off the 1 inside the brackets, 64 the linear like this: right after term, (i..., 1 ) divide .. 8 15 + *(** 4 ㅗ 64 ! 64 step 4: use the three brackets terms to group the fisst inside the brackets .

2 8. 121 64 perfect square the rest term steps using a ²-6² (cath) Carb) s(0-7-3)*) = *(*** ++)(***) = 8(**) (-1) 8 (8x +10) (81-12) ANSWER: 8x2-22--15 = ( 8u+10)(82-12) 0 นาท be the Part(b) What must be square which, when increased by eight of its woots, amounts to 208 AL solution: 2 this eq" The equation ad-Khwarizmi wanted to solne u +84 = 20. solve we probably would by subtracting 20 from both sides then factoring. AL- Khwarizmi instead used the technique of completing the

add we 2 to both the equation square. To complete the square, (9)?= 4= 16 sides of obtaining lige notebbe-elo death 2. x + 8u+16=20+ 16 The left hand side becomes a perfect thus write as (x+4)= 9 = 36. square and we can To solve the equation, we take the square root of each side, 07 obtaining x + 4 = 6 Int 4 = -6 soo u=2 2= 10. Al-Khwasizmi did not consider the Negative His solution was 26=2. since, the solution : original he also asked for the problem Noted square, 2 that x 4.