Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:


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1 Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you to find the solutions of equtions. However, qudrtics re not lwys esy to fctor. Sometimes qudrtics cnnot e fctored completely. Two procedures tht cn help you to fctor qudrtic re completing the squre nd the qudrtic formul. Completing the squre In Appendi A, two specil cses of epnding rckets were considered: ( + ) = + + ( ) = +. These were clled perfect squres. In Appendi C, two strtegies suggested for fctoring lgeric epressions were to look for perfect squres nd to look for differences of squres. Emple D.1 Fctor the qudrtic epression: y looking for perfect squres nd differences of squres. Solution: The given lgeric epression is very close to the perfect squre: ( + 3) = So: = ( ) 1 = ( + 3) 1. Since 1 = 1, the lst epression is difference of two squres, nd so: = ( + 3) 1 = ( )( + 3 1) = ( + 4)( ). Emple D.1 is very elorte nd counterintuitive wy to fctor strightforwrd lgeric epression like The process outlined in Emple D.1 hs one gret dvntge over quicker, more intuitive methods: the method of Emple D.1 will work
2 when the numers involved re more complicted nd it is hrder to fctor the qudrtic epression. Emple D. Fctor the qudrtic epression: y looking for perfect squres nd differences of squres. Solution: The perfect squre tht this qudrtic epression is relted to is: ( + 33) = The qudrtic epression tht we hve to fctor is relted to this perfect squre: = ( ) 1081 = ( + 33) This lst lgeric epression is difference of two squres, so: = ( )( ) = ( )( ). The numer ppers ecuse the squre root of 1081 is pproimtely The process of fctoring qudrtic equtions illustrted in Emples D.1 nd D. is clled completing the squre. As indicted, not ll qudrtics cn e completely fctored. The process of completing the squre cn tell you when this is the cse. Emple D.3 Try to fctor the qudrtic epression: y looking for perfect squres nd differences of squres. Solution: Following the pttern set out in Emples D.1 nd D., you could strt y finding perfect squre tht is closely relted to The perfect squre: ( + 3) = is very closely relted to In fct, = ( ) + 1 = ( + 3) + 1.
3 This is where the pttern estlished in Emples D.1 nd D. reks down, ecuse the epression tht we hve is not difference of two squres. The epression: ( + 3) + 1 cnnot e fctored ny further ecuse it is the sum rther thn the difference of two squres. The Qudrtic Formul The process of fctoring qudrtic epression y completing the squre cn e summrized s n lgeric formul. The formul is this: If you re trying to fctor the qudrtic epression: + + c, then the fctors re: c c There is one importnt cvet when using this method to fctor polynomil: ecuse you hve to tke squre root, the quntity: 4c must e greter thn or equl to zero. If 4c is negtive, then it is not possile to fctor the qudrtic epression + + c. Emple D.4 Use the qudrtic formul to fctor the following qudrtic epressions. If it is not possile to fctor ny of the qudrtic epressions, indicte why you think this to e the cse. ) ) c)
4 Solution: ) To fctor , note the similrities etween this epression nd + + c. The correspondence is tht: = 6. = 9. c =. Before plugging into the formul, it is wise to mke sure tht the quntity 4c is greter thn or equl to zero. (If the quntity 4c is negtive, then the qudrtic epression cnnot e fctored, so there will e no sense in trying.) 4c = = = 33. The numer 33 is greter thn or equl to zero, so the qudrtic will fctor. Plugging = 6, = 9 nd c = into the formul gives: = 6( )( ). ) To fctor + + 1, gin note the similrities etween this epression nd + + c. In this cse, the correspondence is tht: = 1. = 1. c = 1. To check whether or not the epression + + 1, cn e fctored, you cn check the sign of 4c. In this cse, 4c = = 1 4 = 3. Since 4c is negtive, the qudrtic epression cnnot e fctored. c) Here the nlysis is just like the previous two cses, ecept tht: = 1. =. c = 1. Checking the quntity 4c gives 4c = = 0. As this is not negtive, the qudrtic epression will fctor. Plugging = 1, = nd c = 1 into the formul gives: ( + 1)( + 1) = ( + 1). An lterntive (nd eqully vlid) wy to fctor this prticulr qudrtic epression would hve een to relize tht ws perfect squre, so tht you cn determine tht = ( + 1) without hving to use the formul.
5 Using Completing the Squre to Otin the Qudrtic Formul You might wonder how the formul for the fctors of the qudrtic epression + + c ws otined. The working shown elow indictes how the process of completing the squre my e used to otin the formul. The working presented elow is quite formidle ecuse it fetures lot of symols, rther thn just concrete numers. Use the eplntory notes for ech step to follow wht is going on. + + c = c + + = c = c + Fctor out the. Mke the epression s much like the perfect squre: + possile. s Fctor the perfect squre nd group the leftover terms together. c = + Simplify the term: 4 =. 4 = + 4c Put the two terms over common 4 denomintor. (See Appendi E.) Relize tht wht you hve is difference of two squres. = + + 4c + 4 4c Use the difference of squres to 4 fctor. = + + 4c + 4c Tke the squre roots of the numertors nd denomintors. c c = Comine like terms.
6 Eercises for Appendi D For Prolems 110, complete the squre for the given qudrtic epressions. 1. y + y u 14u r + 1r t 7t r 4 r w + 3w. For Prolems 1115, fctor the qudrtic epressions (if possile). 11. r + 7r y + 5y e e e t + 13t u + 4u 5.
7 Answers to Eercises for Appendi D 1. (y + 1) ( + ) (u  7) ( + ) (r + ) (  5) (t  3.5t) (r  4) (w + 3/) (r + 3)(r + 4). 1. (y + 1)(y + 3/). 13. e(e + ) (t )(t ). 15. This qudrtic does not fctor. To see this,  4c = 164(1)(5) = 4.
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