TOPIC 7 Quadratic epressions 7.1 Overview Why learn this? How is your algeraic tool kit? Is there some room to epand your skills? As epressions ecome more comple, more power will e needed to manipulate them and to carry out asic skills such as adding, multiplying and factorising. Dealing with quadratic epressions is the fi rst step to higher-level skills. What do you know? 1 THInK List what you know aout quadratic epressions. Use a thinking tool such as a concept map to show your list. 2 PaIr Share what you know with a partner and then with a small group. 3 SHare As a class, create a thinking tool such as a large concept map that shows your class s knowledge of quadratic epressions. Learning sequence 7.1 Overview 7.2 Epanding algeraic epressions 7.3 Factorising epressions with three terms 7.4 Factorising epressions with two or four terms 7.5 Factorising y completing the square 7.6 Mied factorisation 7.7 Review ONLINE ONLY
WaTCH THIS VIdeO The story of mathematics: Adelard of Bath Searchlight Id: eles-1846
7.2 Epanding algeraic epressions Binomial epansion Consider the rectangle of length a + and width c + d shown elow. Its area is equal to (a + )(c + d). The diagram shows that 1a + 2 1c + d2 ac + ad + c + d. c d a ac ad c d factorised form epanded form Epansion of the inomial epression ( + 3)( + 2) can e shown y this area model. Epressed mathematically this is: u 2 u 3 2 2 2 3 3 3 2 6 There are several methods that can e used to epand inomial factors. FOIL method The word FOIL provides us with an acronym for the epansion of a inomial product. F First: multiply the first terms in each racket ( + a)( ) Outer: multiply the two outer terms Inner: multiply the two inner terms Last: multiply the last terms in each racket v 1 + 32 1 + 22 2 + 2 + 3 + 6 2 + 5 + 6 factorised form epanded form v O ( + a)( ) I ( + a)( ) L ( + a)( ) 272 Maths Quest 10 + 10A
WOrKed example 1 Epand each of the following. a ( + 3)( + 2) ( 7)(6 ) THInK If there is a term outside the pair of rackets, epand the rackets and then multiply each term of the epansion y that term. The square of a inomial The epansion of (a + ) 2 can e represented y this area model. a WrITe a 1 Write the epression. a ( + 3)( + 2) 2 Multiply the first terms in each racket, then the outer terms, the inner terms and finally the last terms., 2, 3, 3 2 2 + 2 + 3 + 6 3 Collect like terms. 2 + 5 + 6 1 Write the epression. ( 7)(6 ) 2 Multiply the first terms in each racket, then the outer terms, the inner terms and finally the last terms. 3 Remove the rackets y multiplying each term in the rackets y the term outside the racket. Rememer to change the sign when the term outside the racket is negative. 6,, 7 6, 7 6 2 42 + 7 4 Collect like terms. 2 + 13 42 WOrKed example 2 Epand 3( + 8)( + 2). THInK TI CASIO WrITe 1 Write the epression. 3( + 8)( + 2) 2 Use FOIL to epand the pair of rackets. 3( 2 + 2 + 8 + 16) 3 Collect like terms within the rackets. 3( 2 + 10 + 16) 4 Multiply each of the terms inside the rackets y the term outside the rackets. 3 2 + 30 + 48 a a a a 2 a a a a 2 Topic 7 Quadratic epressions 273
WOrKed example 3 Similarly (a ) 2 a 2 2a + 2. (a + ) 2 a 2 + a + a + 2 a 2 + 2a + 2 This epansion is often memorised. To find the square of a inomial: square the first term multiply the two terms together and then doule them square the last term. Epand and simplify each of the following. a (2 5) 2 3(2 + 7) 2 THInK The difference of two squares WrITe a 1 Write the epression. a (2 5) 2 2 Epand using the rule (a ) 2 a 2 2a + 2. (2) 2 2 2 5 + (5) 2 4 2 20 + 25 1 Write the epression. 3(2 + 7) 2 2 Epand the rackets using the rule (a + ) 2 a 2 + 2a + 2. 3 Multiply every term inside the rackets y the term outside the rackets. WOrKed example 4 TI CASIO 3[(2) 2 + 2 2 7 + (7) 2 ] 3(4 2 + 28 + 49) 12 2 84 147 When a + is multiplied y a (or vice-versa), (a + )(a ) a 2 a + a 2 a 2 2 The epression is called the difference of two squares and is often referred to as DOTS. This result can e memorised as a short cut. Epand and simplify each of the following. a (3 + 1)(3 1) 4(2 7)(2 + 7) THInK TI CASIO WrITe a 1 Write the epression. a (3 + 1)(3 1) 2 Epand using the rule (a + )(a ) a 2 2. (3) 2 (1) 2 9 2 1 274 Maths Quest 10 + 10A
1 Write the epression. 4(2 7)(2 + 7) 2 Epand using the difference of two squares rule. 4[(2) 2 (7) 2 ] 4(4 2 49) 3 Multiply y 4. 16 2 196 Eercise 7.2 Epanding algeraic epressions IndIVIdual PaTHWaYS PraCTISe Questions: 1a f, 2a h, 3a d, 5a d, 6a, 8a d, 9a d, 10a f, 11 14 COnSOlIdaTe Questions: 1d l, 2d j, 3c f, 4a c, 5c f, 6, 7, 8c f, 9c f, 10 17, 19 Individual pathway interactivity int-4596 master Questions: 1d l, 2f l, 3e i, 4, 5e h, 6, 7, 8g l, 9e i, 10 20 FluenCY 1 Epand each of the following. a 2( + 3) 4( 5) c 3(7 ) d ( + 3) e ( + 2) f 2( 4) g 3(5 2) h 5(2 3) i 2(4 + 1) j 2 2 (2 3) k 3 2 (2 1) l 5 2 (3 + 4) 2 WE1 Epand each of the following. a ( + 3)( 4) ( + 1)( 3) c ( 7)( + 2) d ( 1)( 5) e (2 )( + 3) f ( 4)( 2) g (2 3)( 7) h ( 1)(3 + 2) i (3 1)(2 5) j (3 2)(7 ) k (5 2)(3 + 4) l (11 3)(10 + 7) 3 WE2 Epand each of the following. a 2( + 1)( 3) 4(2 + 1)( 4) c 2( + 1)( 7) d 2( 1)( + 1) e 3( 5)( + 5) f 6( 3)( + 3) g 2(3 )( 3) h 5(2 )( 4) i 6( + 5)(4 ) 4 Epand each of the following. a ( 1)( + 1)( + 2) ( 3)( 1)( + 2) c ( 5)( + 1)( 1) d ( 1)( 2)( 3) e (2 1)( + 1)( 4) f (3 + 1)(2 1)( 1) 5 Epand each of the following and simplify. a ( + 2)( 1) 2 3 (2 5)( + 2) c (2 3)( + 1) + (3 + 1)( 2) d (3 2)(2 1) + (4 5)( + 4) e ( + 1)( 7) ( + 2)( 3) f ( 2)( 5) ( 1)( 4) g ( 3)( + 1) +!3 h (!2 3)(!3 + 2)!5 6 MC a (3 1)(2 + 4) epands to: a 6 2 + 10 4 5 2 24 + 3 C 3 2 + 2 4 d 6 2 10 4 e 6 2 4 reflection Why does the difference of two squares rule have that name? doc-5244 doc-5245 Topic 7 Quadratic epressions 275
2( 1)( + 3) epands to: A 2 + 2 3 B 2 2 4 + 6 C 2 3 4 2 + 6 D 2 3 + 4 2 6 E 2 3 3 7 MC The epression ( 1)( 3)( + 2) is not the same as: A ( 3)( 1)( + 2) B ( + 3)( 1)( 2) C ( 1)( + 2)( 3) D ( + 2)( 1)( 3) E ( 3)( + 2)( 1) 8 WE3a Epand and simplify each of the following. a ( 1) 2 ( + 2) 2 c ( + 5) 2 d (4 + ) 2 e (7 ) 2 f (12 ) 2 g (3 1) 2 h (12 3) 2 i (5 + 2) 2 j (2 3) 2 k (5 4) 2 l (1 5) 2 9 WE3 Epand and simplify each of the following. a 2( 3) 2 4( 7) 2 c 3( + 1) 2 d (2 + 3) 2 e (7 1) 2 f 2(2 3) 2 g 3(2 9) 2 h 5(3 11) 2 i 4(2 + 1) 2 10 WE4 Epand and simplify each of the following. a ( + 7)( 7) ( + 9)( 9) c ( 5)( + 5) d ( 1)( + 1) e (2 3)(2 + 3) f (3 1)(3 + 1) g (7 )(7 + ) h (8 + )(8 ) i (3 2)(3 + 2) UNDERSTANDING 11 The length of the side of a rectangle is ( + 1) cm and the width is ( 3) cm. a Find an epression for the area of the rectangle. Simplify the epression y epanding. c If 5 cm, find the dimensions of the rectangle and, hence, its area. 12 Chickens are kept in a square enclosure with sides measuring m. The numer of chickens is increasing and so the size of the enclosure is to have 1 metre added to one side and 2 metres to the adjacent side. a Draw a diagram of the original enclosure. Add to the first diagram or draw another one to show the new enclosure. Mark the lengths on each side on your diagram. c Write an epression for the area of the new enclosure in factorised form. d Epand and simplify the epression y removing the rackets. e If the original enclosure had sides of 2 metres, find the area of the original square and then the area of the new enclosure. 13 Shown elow are three students attempts at epanding (3 + 4)(2 + 5). STUDENT A STUDENT B 276 Maths Quest 10 + 10A
STUDENT C a Which student s work was correct? Copy each of the incorrect answers into your workook and correct the mistakes in each as though you were the teacher of these students. 14 If a 5 and 3, show that (a )(a + ) a 2 2 y evaluating oth epressions. 15 If a 5 and 3, show that (a + ) 2 a 2 + 2a + 2 y evaluating oth epressions. 16 Write an epression in factorised and epanded form that is: a a quadratic trinomial the square of a inomial c the difference of two squares d oth a and. REASONING 17 Eplain the difference etween the square of a inomial and the difference etween two squares. 18 Show that (a + )(c + d) (c + d)(a + ). PROlem SOLVING 19 Epand: a (2 + 3y 5z) 2 20 Find an epanded epression for: a the volume of the cuoid ( 2) cm (3 4) cm the total surface area of the square-ased pyramid. aa1 + 1 2 22 (2 3) cm (2 1) cm (2 1) cm (2 + 3) cm (2 + 3) cm Topic 7 Quadratic epressions 277
7.3 Factorising epressions with three terms A monic quadratic epression is an epression in the form a 2 + + c where a 1. Factorising monic quadratic trinomials The area model of inomial epansion can e used to find a pattern for factorising a general quadratic epression. For eample, ( + f)( + h) 2 + f + h + fh ( + 4)( + 3) 2 + 4 + 3 + 12 2 + (f + h) + fh 2 + 7 + 12 + h 2 h + f f fh To factorise a general quadratic, look for factors of c that add to. 2 + + c ( + f)( + h) Factors of c that add to 3 + 4 7 For eample, 2 + 7 + 12 ( + 3)( + 4). 3 4 12 + 3 2 3 + 4 4 12 278 Maths Quest 10 + 10A
WOrKed example 5 Factorise the following quadratic epressions. a 2 + 5 + 6 2 + 10 + 24 THInK a 1 Write the epression and: i check for a common factor. There is no common factor. ii check for a DOTS pattern. The epression is not in the form a 2 2. iii check for a perfect squares pattern. 6 is not a perfect square. This must e a general quadratic epression. 2 i The general quadratic epression has the pattern 2 + 5 + 6 ( + f)( + h). f and h are a factor pair of 6 that add to 5. ii Calculate the sums of factor pairs of 6. The factors of 6 that add to 5 are 2 and 3, as shown in lue. 3 Sustitute the values of f and h into the epression in its factorised form. 1 Check for patterns of common factors, DOTS and perfect squares patterns. None of these apply, so the epression is a general quadratic. 2 i The general quadratic epression has the pattern 2 + 10 + 24 ( + f)( + h), where f and h are a factor pair of 24 that add to 10. ii Calculate the sums of factor pairs of 24. The factors of 24 that add to 10 are 4 and 6, as shown in lue. 3 Sustitute the values of f and h into the epression in its factorised form. Factorising non-monic quadratic trinomials A non-monic quadratic epression is a 2 + + c where a 1. When a quadratic trinomial in the form a 2 + + c is written as a 2 + m + n + c, where m + n, the four terms can e factorised y grouping. 2 2 + 11 + 12 2 2 + 8 + 3 + 12 2 2 + 8 + 3 + 12 2( + 4) + 3( + 4) 2( + 4) + 3( + 4) ( + 4)(2 + 3) WrITe a 2 + 5 + 6 Factors of 6 Sum of factors 1 and 6 7 2 and 3 5 2 + 5 + 6 ( + 2)( + 3) 2 + 10 + 24 Factors of 24 Sum of factors 1 and 24 25 2 and 12 14 3 and 8 11 4 and 6 10 2 + 10 + 24 ( + 4)( + 6) Topic 7 Quadratic epressions 279
There are many cominations of numers that satisfy m + n ; however, only one particular comination can e grouped and factorised. For eample, 2 2 + 11 + 12 2 2 + 7 + 4 + 12 or 2 2 + 11 + 12 2 2 + 8 + 3 + 12 2 2 + 7 + 4 + 12 2 2 + 8 + 3 + 12 (2 + 7) + 4( + 3) 2( + 4) + 3( + 4) cannot e factorised further ( + 4) + (2 + 3) In eamining the general inomial epansion, a pattern emerges that can e used to help identify which comination to use for m + n. (d + e)(f + g) df 2 + dg + ef + eg df 2 + (dg + ef) + eg m + n dg + ef and m n dg ef dgef dfeg ac Therefore, m and n are factors of ac that sum to. To factorise a general quadratic where a 1, look for factors of ac that sum to. Then rewrite the quadratic trinomial with four terms that can then e grouped and factorised. WOrKed example 6 a 2 + + c a 2 + m + n + c Factors of ac that sum to Factorise 6 2 11 10. THInK 1 Write the epression and look for common factors and special patterns. The epression is a general quadratic with a 6, 11 and c 10. 2 Since a 1, rewrite a 2 + + c as a 2 + m + n + c, where m and n are factors of ac (6 10) that sum to ( 11). Calculate the sums of factor pairs of 60. As shown in lue, 4 and 15 are factors of 60 that add to 11. 3 Rewrite the quadratic epression: a 2 + + c a 2 + m + n + c with m 4 and n 15. 4 Factorise using the grouping method: 6 2 + 4 2(3 + 2) and 15 10 5 (3 + 2) Write the answer. TI CASIO WrITe 6 2 11 10 6 2 + 11 + 10 Factors of 60 (6 10) Sum of factors 60, 1 59 20, 3 17 30, 2 28 15, 4 11 15, 4 11 6 2 11 10 6 2 + 4 + 15 10 6 2 11 10 2(3 + 2) + 5(3 + 2) (3 + 2)(2 5) 280 Maths Quest 10 + 10A
Eercise 7.3 Factorising epressions with three terms IndIVIdual PaTHWaYS PraCTISe Questions: 1a e, 2a e, 3a e, 4 6, 7a e, 8, 11, 15 COnSOlIdaTe Questions: 1f j, 2f i, 3f i, 4, 5, 6, 7d h, 8, 9a d, 11, 12, 15, 16 Individual pathway interactivity int-4597 master Questions: 1k o, 2k l, 3j l, 4, 5, 6, 7i l, 8, 9, 10, 13 17 FluenCY 1 WE5 Factorise each of the following. a 2 + 3 + 2 2 + 4 + 3 c 2 + 10 + 16 d 2 + 8 + 16 e 2 2 3 f 2 3 4 g 2 11 12 h 2 4 12 i 2 + 3 4 j 2 + 4 5 k 2 + 6 7 l 2 + 3 10 m 2 4 + 3 n 2 9 + 20 o 2 + 9 70 2 WE6 Factorise each of the following. a 2 2 20 18 3 2 9 6 c 2 3 2 d 2 11 10 e 2 7 10 f 2 13 12 g 2 7 12 h 2 8 12 i 2 2 + 14 + 20 j 3 2 + 33 + 30 k 5 2 + 105 + 100 l 5 2 + 45 + 100 3 Factorise each of the following. a a 2 6a 7 t 2 6t + 8 c 2 + 5 + 4 d m 2 + 2m 15 e p 2 13p 48 f c 2 + 13c 48 g k 2 + 22k + 57 h s 2 16s 57 i g 2 g 72 j v 2 28v + 75 k 2 + 14 32 l 2 19 + 60 4 MC a To factorise 14 2 49 + 21, the first step is to: a find factors of 14 and 21 that will add to 49 take out 14 as a common factor C take out 7 as a common factor d find factors of 14 and 49 that will add to make 21 e take out 14 as a common factor The epression 42 2 9 6 can e completely factorised to: a (6 3)(7 + 2) 3(2 1)(7 + 2) C (2 1)(21 + 6) d 3(2 + 1)(7 2) e 42( 3)( + 2) 5 MC When factorised, ( + 2) 2 (y + 3) 2 equals: a ( + y 2)( + y + 2) ( y 1)( + y 1) C ( y 1)( + y + 5) d ( y + 1)( + y + 5) e ( + y 1)( + y + 2) 6 Factorise each of the following using an appropriate method. a 2 2 + 5 + 2 2 2 3 + 1 c 4 2 17 15 d 4 2 + 4 3 e 2 2 9 35 f 3 2 + 10 + 3 reflection In your own words, descrie how you would factorise a quadratic trinomial. doc-5250 Topic 7 Quadratic epressions 281
g 6 2 17 + 7 h 12 2 13 14 i 10 2 9 9 j 20 2 + 3 2 k 12 2 + 5 2 l 15 2 + 2 7 Factorise each of the following, rememering to look for a common factor first. a 4 2 + 2 6 9 2 60 21 c 72 2 + 12 12 d 18 2 + 3 + 3 e 60 2 + 150 + 90 f 24a 2 + 18a 105a g 8 2 + 22 12 h 10 2 + 31 + 14 i 24 2 + 35 4 j 12 2 2y + 2y 2 k 30 2 + 85y + 70y 2 l 600 2 780y 252y 2 UNDERSTANDING 8 Consider the epression ( 1) 2 + 5( 1) 6. a Sustitute w 1 in this epression. Factorise the resulting quadratic. c Replace w with 1 and simplify each factor. This is the factorised form of the original epression. 9 Use the method outlined in question 8 to factorise each of the following epressions. a ( + 1) 2 + 3( + 1) 4 ( + 2) 2 + ( + 2) 6 c ( 3) 2 + 4( 3) + 4 d ( + 3) 2 + 8( + 3) + 12 e ( 7) 2 7( 7) 8 f ( 5) 2 3( 5) 10 10 Factorise 2 + 0.75. 11 Students decide to make Valentine s Day cards. The total area of each card is equal to ( 2 4 5) cm 2. a Factorise the epression to find the dimensions of the cards in terms of. Write down the length of the shorter side in terms of. c If the shorter sides of a card are 10 cm in length and the longer sides are 16 cm in length, find the value of. d Find the area of the card proposed in part c. Happy e If the students want to make 3000 Valentine s Day cards, how Valentine s much cardoard will e required? Give your answer in terms of. Day 12 The area of a rectangular playground is given y the general epression (6 2 + 11 + 3) m 2 where is a positive whole numer. a Find the length and width of the playground in terms of. Write an epression for the perimeter of the playground. c If the perimeter of a particular playground is 88 metres, find. REASONING 13 Cameron wants to uild an in-ground endless pool. Basic models have a depth of 2 metres and a length triple the width. A spa will also e attached to the end of the pool. a The pool needs to e tiled. Write an epression for the surface area of the empty pool (that is, the floor and walls only). 282 Maths Quest 10 + 10A
The spa needs an additional 16 m 2 of tiles. Write an epression for the total area of tiles needed for oth the pool and the spa. c Factorise this epression. d Cameron decides to use tiles that are selling at a discount price, ut there are only 280 m 2 of the tile availale. Find the maimum dimensions of the pool he can uild if the width is in whole metres. Assume the spa is to e included in the tiling. e What area of tiles is actually needed to construct the spa and pool? f What volume of water can the pool hold? 14 Faric pieces comprising yellow squares, white squares and lack rectangles are sewn together to make larger squares (patches) as shown in the diagram. The length of each lack rectangle is twice its width. These patches are then sewn together to make a patchwork quilt. A finished square quilt, made from 100 patches, has an area of 1.44 m 2. a Determine the size of each yellow, lack and white section in one faric piece. Show your working. How much (in m 2 ) of each of the coloured farics would e needed to construct the quilt? (Ignore seam allowances.) c Sketch a section of the finished product. y y w y y 15 Each factorisation elow contains an error. Identify the error in each statement. a 2 7 + 12 ( + 3)( 4) 2 12 ( 3)( + 4) c 2 + 2 ( 1)( + 1) d 2 4 21 ( 3)( 7) e 2 + 4 21 ( + 3)( 7) f 2 30 ( 5)( + 6) g 2 + 7 8 ( + 1)( 8) h 2 11 + 30 ( 5)( + 6) PrOlem SOlVIng 16 Factorise: a 6(3a 1) 2 13(3a 1) 5 3m 4 19m 2 14 c 2 sin 2 () 3 sin() + 1. 17 Factorise: a 2 2 + 3!3 6 (z + 1) 3 + (z 1) 3. 7.4 Factorising epressions with two or four terms Factorising epressions with two terms If the terms in an epanded epression have a common factor, the highest common factor is written at the front of the rackets and the remaining factor for each term in the epression is written in the rackets. For eample, 4 2 36 4( 2 9). A Difference of Two Squares (DOTS) epression in epanded form has two squared terms separated y a sutraction symol. a 2 2 (a )(a + ) Epanded form Factorised form doc-5251 Topic 7 Quadratic epressions 283
WOrKed example 7 Factorise the following. a 12k 2 + 18 16a 2 25 4 THInK a 1 Write the epression and look for common factors. The terms have a highest common factor of 6. Write the 6 in front of a set of rackets, then determine what must go inside the rackets. 12k 2 6 2k 2, 18 6 3 2 Look for patterns in the epression inside the rackets to factorise further. The epression inside the rackets cannot e factorised further. 1 Write the epression and look for common factors. The epression has no common factor. 2 Look for the DOTS pattern in the epression. Write the equation showing squares. 3 Use the pattern for DOTS to write the factors. a 2 2 (a + )(a ) Factorising epressions with four terms If there are four terms to e factorised, look for a common factor first. Then group the terms in pairs and look for a common factor in each pair. It may e that a new common factor emerges as a racket (common inomial factor). If an epression has four terms, it may require grouping to factorise it. In the process known as grouping two and two, the terms of the epression are grouped into two pairs, then a common factor is removed from each pair. When selecting terms to place as pairs, each pair after factorising should result in a common inomial factor. For eample: WOrKed example 8 Factorise each of the following. a 4y + m 4my THInK 2a 6 + 3ac 9c 2(a 3) + 3c(a 3) a 1 Write the epression and look for a common factor. (There isn t one.) 2 Group the terms so that those with common factors are net to each other. TI CASIO 3 Take out a common factor from each group (it may e 1). 4 Factorise y taking out a common inomial factor. The factor ( 4y) is common to oth groups. WrITe 2 + 3 y 2 + 3y a 12k 2 + 18 6(2k 2 + 3) 16a 2 25 4 WrITe a 4 2 a 2 5 2 ( 2 ) 2 (4a) 2 (5 2 ) 2 (4a + 5 2 )(4a 5 2 ) 4y + m 4my ( 4y) + (m 4my) 1( 4y) + m( 4y) ( 4y)(1 + m) 284 Maths Quest 10 + 10A
1 Write the epression and look for a common factor. 2 Group the terms so that those with common factors are net to each other. Now we will look at grouping a different comination, known as grouping three and one. Eercise 7.4 Factorising epressions with two or four terms IndIVIdual PaTHWaYS PraCTISe Questions: 1 4, 8a h, 9a d, 10a c, 14, 15 COnSOlIdaTe Questions: 1 4, 5a, 7a c, 8a h, 9a d, 10a d, 11 13, 15, 16, 18 Individual pathway interactivity int-4598 2 + 3 y 2 + 3y ( 2 y 2 ) + (3 + 3y) 3 Factorise each group. ( + y)( y) + 3( + y) 4 Factorise y taking out a common inomial factor. The factor ( + y) is common to oth groups. WOrKed example 9 Factorise the following epression: 2 + 12 + 36 y 2. THInK 1 Write the epression and look for a common factor. 2 Group the terms so that those that can e factorised are net to each other. 3 Factorise the quadratic trinomial. This is the form of a perfect square. 4 Factorise the epression using a 2 2 (a + )(a ). TI CASIO ( + y)( y + 3) WrITe 2 + 12 + 36 y 2 ( 2 + 12 + 36) y 2 ( + 6)( + 6) y 2 ( + 6) 2 y 2 ( + 6 + y)( + 6 y) master Questions: 1 7, 8e l, 9 19 reflection What do you always check for fi rst when factorising? FluenCY 1 Factorise each of the following y taking out a common factor. a 2 + 3 2 4 c 3 2 6 d 4 2 + 16 e 9 2 3 f 8 8 2 g 12 3 2 h 8 12 2 i 8 2 11 Topic 7 Quadratic epressions 285
doc-5246 doc-5247 2 Factorise each of the following y taking out a common inomial factor. a 3( 2) + 2( 2) 5( + 3) 2( + 3) c ( 1) 2 + 6( 1) d ( + 1) 2 2( + 1) e ( + 4)( 4) + 2( + 4) f 7( 3) ( + 3)( 3) 3 WE7 Factorise each of the following. a 2 1 2 9 c 2 25 d 2 100 e y 2 k 2 f 4 2 9y 2 g 16a 2 49 h 25p 2 36q 2 i 1 100d 2 4 Factorise each of the following. a 4 2 4 5 2 80 c a 2 9a d 2 2 8d 2 e 100 2 1600 f 3a 2 147a g 4p 2 256p h 36 2 16 i 108 3 2 5 MC a If the factorised epression is ( + 7)( 7), then the epanded epression must have een: a 2 7 2 + 7 C 2 49 d 2 + 49 e 2 14 + 49 If the factorised epression is a 4 3 5 a 4 + 3, then the original epression must have een: 5 a 2 4 3 5 2 16 9 25 C 2 4 (!3)2 (!5) 2 d 2 4 9 25 c The factorised form of 64 2 9y 2 is: a (64 + 9y)(64 9y) (8 + 3y)(8 3y) C (8 3y)(8 3y) d (8 + 3y)(8 + 3y) e (16 + 3y)(16 3y) 6 MC Which of the following epressions would e factorised y grouping two and two? a 2 a 2 + 12a 36 2 7 10 C 2 2 6 y + 3y d (s 5) 2 25(s + 3) 2 e (r + 5) (r + 3)(r + 5) 7 Factorise each of the following over the set of real numers. a 2 11 2 7 c 2 15 d 4 2 13 e 9 2 19 f 3 2 66 g 5 2 15 h 2 2 4 i 12 2 36 8 Factorise each of the following epressions. a ( 1) 2 4 ( + 1) 2 25 c ( 2) 2 9 d ( + 3) 2 16 e 49 ( + 1) 2 f 36 ( 4) 2 g ( 1) 2 ( 5) 2 h 4( + 2) 2 9( 1) 2 i 25( 2) 2 16( + 3) 2 9 WE8a Factorise each of the following. a 2y + a 2ay 2 + a + 2y + ay c a ay + y d 4 + 4y + z + yz e ef 2e + 3f 6 f mn 7m + n 7 e 2 16 (!3)2 (!5) 2 286 Maths Quest 10 + 10A
g 6rt 3st + 6ru 3su h 7mn 21n + 35m 105 i 64 8j + 16k 2jk j 3a 2 a 2 + 3ac ac k 5 2 + 10 + 2 y + 2y l 2m 2 m 2 n + 2mn mn 2 10 Factorise each of the following. a y + 7 2y 14 mn + 2n 3m 6 c pq + 5p 3q 15 d s 2 + 3s 4st 12t e a 2 cd c + a 2 d f y z 5z 2 + 5yz 11 WE8 Factorise each of the following. a a 2 2 + 4a 4 p 2 q 2 3p + 3q c m 2 n 2 + lm + ln d 7 + 7y + 2 y 2 e 5p 10pq + 1 4q 2 12 WE9 Factorise each of the following. f 49g 2 36h 2 28g 24h a 2 + 14 + 49 y 2 2 + 20 + 100 y 2 c a 2 22a + 121 2 d 9a 2 + 12a + 4 2 e 25p 2 40p + 16 9t 2 f 36t 2 12t + 1 5v 2 13 MC a In the epression 3( 2) + 4y( 2), the common inomial factor is: A 3 + 4y B 3 4y C D + 2 E 2 Which of the following terms is a perfect square? A 9 B ( + 1)( 1) C 3 2 D 5(a + ) 2 E 25 c Which of the following epressions can e factorised using grouping? A 2 y 2 B 1 + 4y 2y + 4 2 C 3a 2 + 8a + 4 D 2 + + y y 2 E 2a + 4 6a + 18 14 MC When factorised, 6(a + ) (a + ) equals: A 6 (a + ) B (6 )(a + ) C 6(a + ) D (6 + )(a ) E (6 + )(a + ) UNDERSTANDING 15 The area of a rectangle is ( 2 25) cm 2. a Factorise the epression. Find the length of the rectangle if the width is + 5 cm. c If 7 cm, find the dimensions of the rectangle. d Hence, find the area of the rectangle. e If 13 cm, how much igger would the area of this rectangle e? REASONING 16 A circular garden of diameter 2r m is to have a gravel path laid around it. The path is to e 1 m wide. a Find the area of the garden in terms of r. Find the area of the garden and path together in terms of r, using the formula for the area of a circle. Topic 7 Quadratic epressions 287
c Write an epression for the area of the path in fully factorised form. d If the radius of the garden is 5 m, then find the area of the path, correct to 2 decimal places. Show your working. 17 A roll of material is ( + 2) metres wide. Annie uys ( + 3) metres of the material and Bronwyn uys 5 metres of the material. a Write an epression, in terms of, for the area of each piece of material purchased. If Annie has ought more material than Bronwyn, write an epression for how much more she has than Bronwyn. c Factorise and simplify this epression. d Find the width of the material if Annie has 5 m 2 more than Bronwyn. e How much material does each person have? Eplain your answer. PROlem SOLVING A polynomial in the form a 3 3 is known as the difference of two cues. The difference of two cues can e factorised as: a 3 3 (a )(a 2 + a + 2 ) Use either the difference of two squares or the difference of two cues to answer these prolem solving questions. 18 Factorise: a 2 4y + 4y 2 a 2 + 6a 9 2 3 + 9 2 + 27 37. 19 Factorise: a 27 3 1 12 2 75y 2 9(4 3). 288 Maths Quest 10 + 10A
7.5 Factorising y completing the square Completing the square Completing the square is the process of writing a general quadratic epression in turning point form. int-2783 a 2 + + c a( h) 2 + k General form Turning point form The epression 2 + 8 can e modelled as a square with a smaller square missing from the corner, as shown elow. 2 + 2 + 8 Complete the square 8 8 + 4 2 4 + 4 4 ( + 4) 2 (4) 2 In completing the square, the general equation is written as the area of the large square minus the area of the small square. In general, to complete the square for 2 +, the small square has a side length equal to half of the coefficient of ; that is, the area of the small square is a 2 2. 2 + + 2 2 2 + 2 2 4 4 2 + ( ) ( ) 2 + 2 2 2 2 2 Topic 7 Quadratic epressions 289
WOrKed example 10 Write the following in turning point form y completing the square. a 2 + 4 2 + 7 + 1 THInK a 1 The square will consist of a square that has an area of 2 and two identical rectangles with a total area of 4. The length of the large square is ( + 2) so its area is ( + 2) 2. The area of the smaller square is (2) 2. Write 2 + 4 in turning point form. Complete the square with the terms containing. The square will consist of a square that has an area of 2 and two identical rectangles with a total area of 7. The length of the large square is a + 7 2 so its area is a + 7 2 2. The area of the smaller square is a 7 2 2. Write 2 + 7 + 1 in turning point form. WrITe The process of completing the square is sometimes descried as the process of adding the square of half of the coefficient of then sutracting it, as shown in green elow. The result of this process is a perfect square that is then factorised, as shown in lue. 2 + 2 + + a 2 2 a 2 2 a + 2 2 2 + 2 2 2 2 2 + 4 ( + 2) 2 (2) 2 ( + 2) 2 4 2 Simplify the last two terms. a + 7 2 2 49 + 7 2 2 7 2 7 + 2 7 2 ( ) 7 2 2 2 + 7 + 1 a + 7 2 2 a 7 2 2 + 1 a + 7 2 2 45 4 + 4 4 4 2 + + a 2 2 a 2 2 a + 2 2 a 2 2 290 Maths Quest 10 + 10A
For eample, factorise 2 + 8 + 2 y completing the square. 2 + 8 + a 8 2 2 a 8 2 2 + 2 2 + 8 + (4) 2 (4) 2 + 2 2 + 8 + 16 16 + 2 ( + 4) 2 14 Factorising y completing the square When an equation is written in turning point form, it can e factorised as a difference of two squares. WOrKed example 11 Factorise the following y completing the square. a 2 + 4 + 2 2 9 + 1 THInK a 1 To complete the square, add the square of half of the coefficient of and then sutract it. 2 Write the perfect square created in its factorised form. 3 Write the epression as a difference of two squares y: simplifying the numerical terms writing the numerical term as a square 12 1!22 2 2. 4 Use the pattern for DOTS, a 2 2 (a )(a + ), where a ( + 2) and!2. 1 To complete the square, add the square of half of the coefficient of, then sutract it. TI CASIO WrITe a 2 + 4 + 2 2 + 4 + a 4 2 2 a 4 2 2 + 2 2 + 4 + (2) 2 (2) 2 + 2 ( + 2) 2 (2) 2 + 2 ( + 2) 2 4 + 2 ( + 2) 2 2 ( + 2) 2 (!2) 2 1 + 2 +!22 1 + 2!22 2 9 + 1 2 9 + a 9 2 2 a 9 2 2 + 1 2 Write the perfect square created in its factorised form. 2 9 + Q 9 2 R2 Q 9 2 R2 + 1 Q 9 2 R2 Q 9 2 R2 + 1 Topic 7 Quadratic epressions 291
reflection Why is this method called completing the square? 3 Write the epression as a difference of two squares y: simplifying the numerical terms writing the numerical term as a square. 77 4 Q # 77 4 R2 Q!77 2 R2 4 Use the pattern for DOTS: a 2 2 (a )(a + ), where a a + 9 and 2!77 2. Rememer that you can epand the rackets to check your answer. If the coefficient of 2 1, factorise the epression efore completing the square. Eercise 7.5 Factorising y completing the square IndIVIdual PaTHWaYS PraCTISe Questions: 1a d, 2a d, 3a d, 4a d, 5 7, 9 Q 9 2 R2 81 4 + 1 Q 9 2 R2 77 4 Q 9 2 R2 Q!77 2 R2 1 9 +!77 2 1 9!77 2 2 2 COnSOlIdaTe Questions: 1e i, 2e h, 3e h, 4e h, 5 8, 10 Individual pathway interactivity int-4599 master Questions: 1g i, 2g i, 3g i, 4g i, 5 11 FluenCY 1 WE10 Complete the square for each of the following epressions. a 2 + 10 2 + 6 c 2 4 d 2 + 16 e 2 20 f 2 + 8 g 2 14 h 2 + 50 i 2 + 7 j 2 2 WE11a Factorise each of the following y first completing the square. a 2 4 7 2 + 2 2 c 2 10 + 12 d 2 + 6 10 e 2 + 16 1 f 2 14 + 43 g 2 + 8 + 9 h 2 4 13 i 2 12 + 25 3 WE11 Factorise each of the following y first completing the square. a 2 1 2 3 3 c 2 + 5 d 2 + 3 1 e 2 + 5 + 2 f 2 + 5 2 g 2 7 1 h 2 9 + 13 i 2 3 4 Factorise each of the following y first looking for a common factor and then completing the square. a 2 2 + 4 4 4 2 8 20 c 5 2 + 30 + 5 d 3 2 12 39 e 5 2 30 + 10 f 6 2 + 24 6 g 3 2 + 30 + 39 h 2 2 8 14 i 6 2 + 36 30 2 2 292 Maths Quest 10 + 10A
UNDERSTANDING 5 Which method of factorising is the most appropriate for each of the following epressions? a Factorising using common factors Factorising using the difference of two squares rule c Factorising y grouping d Factorising quadratic trinomials e Completing the square i 3 2 8 3 ii 49m 2 16n 2 iii 2 + 8 + 4 y 2 iv 7 2 28 v 6a 6 + a 2 2 vi 2 + 5 vii ( 3) 2 + 3( 3) 10 viii 2 7 1 6 MC a To complete the square, the term which should e added to 2 + 4 is: A 16 B 4 C 4 D 2 E 2 To factorise the epression 2 3 + 1, the term that must e oth added and sutracted is: A 9 B 3 C 3 D 3 E 9 2 4 7 MC The factorised form of 2 6 + 2 is: A 1 + 3!72 1 + 3 +!72 B 1 + 3!72 1 3 +!72 C 1 3!72 1 3!72 D 1 3!72 1 + 3 +!72 E 1 3 +!72 1 3!72 REASONING 8 A square measuring cm in side length has a cm added to its length and cm added to its width. The resulting rectangle has an area of ( 2 + 6 + 3) cm 2. Find the values of a and, correct to 2 decimal places. 9 Show that 2 + 4 + 6 cannot e factorised y completing the square. PROlem SOLVING 10 For each of the following, complete the square to factorise the epression. a 2 2 + 8 + 1 3 2 7 + 5 11 Use the technique of completion of the square to factorise 2 + 2(1 p) + p(p 2). 7.6 Mied factorisation Apply what has een covered in this chapter to the following eercise. Factorising monic and non-monic trinomials: factorising y grouping difference of two squares completing the square. Topic 7 Quadratic epressions 293
reflection When an epression is fully factorised, what should it look like? doc-5248 doc-5249 doc-5252 Eercise 7.6 Mied factorisation IndIVIdual PaTHWaYS PraCTISe Questions: 1 6, 8 10, 12 15, 18, 20, 21, 25, 26, 31 35, 46, 48 COnSOlIdaTe Questions: 1, 2, 4 9, 11, 13 16, 18 20, 22, 26 29, 30, 34, 36, 38, 40, 42, 44, 46, 47a g, 48, 49, 52 FluenCY Factorise each of the following epressions in questions 1 45. 1 3 + 9 2 2 + 4 + 4 9y 2 3 2 36 4 2 49 5 5 2 9 2 6 15 20y 7 5c + de + dc + 5e 8 5 2 80 9 2 6 5 10 2 + 12 11 mn + 1 + m + n 12 2 7 13 16 2 4 14 5 2 + 60 + 100 15 18 + 9 6y 3y 16 2 8 + 16 y 2 17 4 2 + 8 18 fg + 2h + 2g + f h 19 2 5 20 10mn 5n + 10m 5 21 2 + 6 + 5 22 2 10 11 23 2 4 24 5a + c + ac 5 25 y 1 + y 26 3 2 + 5 + 2 27 7 2 28 28 4 2 28 24 29 2p rs + pr 2s 30 3 2 27 31 3u + tv + ut 3v 32 2 11 33 12 2 7 + 1 34 ( 1) 2 4 35 ( + 2) 2 16 36 (2 + 3) 2 25 37 3( + 5) 2 27 38 25 ( 2) 2 39 4(3 ) 2 16y 2 40 ( + 2y) 2 (2 + y) 2 41 ( + 3) 2 ( + 1) 2 42 (2 3y) 2 ( y) 2 master Questions: 1 9, 13 15, 17, 18, 20, 23, 24, 26, 34 37, 39, 41, 43, 45, 47h j, 48, 50 53 43 ( + 3) 2 + 5( + 3) + 4 44 ( 3) 2 + 3( 3) 10 45 2( + 1) 2 + 5( + 1) + 2 understanding 46 Consider the following product of algeraic fractions. 2 + 3 10 2 4 Individual pathway interactivity int-4600 2 + 4 + 4 2 2 8 a Factorise the epression in each numerator and denominator. Cancel factors common to oth the numerator and the denominator. c Simplify the epression as a single fraction. 294 Maths Quest 10 + 10A
47 Use the procedure in question 46 to factorise and simplify each of the following. a 2 4 + 3 2 4 12 2 + 5 + 6 2 9 32 17 + 10 6 2 + 5 6 2 1 2 6 + 5 c d 6 12 2 4 3 + 6 ( 5) 6 2 2 2 2 + 3 + 1 22 + 1 3 2 + 10 8 e 2 + 4 5 2 + 2 2 + 10 + 25 2 + 4 + 4 f g 2 7 + 6 2 + 2 2 12 2 2 8 4a + 8a 5ac + 5a (c 3) c 2 2c 3 h p2 7p p 2 49 p2 + p 6 p 2 + 14p + 49 i j m 2 + 4m + 4 n 2 4m 2 4m 15 2m2 + 4m 2mn 10m 2 + 15m d 2 6d + 9 25e 2 4d 12 20e 4d 2 5d 6 15d 10 48 Find the original epression if the factorised epression is a 4 + 3 5 a 4 3 5. 49 Factorise the following using grouping three and one and DOTS. a 2 18 + 81 y 2 4 2 + 12 16y 2 + 9 REASONING 50 Epand the inomial factors ( + a)( a) and ( + a)( a). What do you notice? Use your findings to factorise each of the following, giving two possile answers. a 2 169 36 2 144c 2 c 225 4 y 2 169 2 y 6 51 Use grouping two and two and DOTS to factorise the following. Show your working. a 2 + 3 y 2 + 3y 7 + 7y + 2 y 2 c 5p 10pq + 1 4q 2 Topic 7 Quadratic epressions 295
NUMBER AND ALGEBRA PROBLEM SOLVING 52 Simplify 2a2 7a + 6 5a2 + 11a + 2 10a2 13a 3. a3 + 8 a3 8 a2 2a + 4 Note: Use the difference of two cues (page 286). doc-5254 53 Factorise 2 + 12 + 40 42y2 4. U N C O R R EC TE D PA G E PR O O FS COMMUNICATING 296 Maths Quest 10 + 10A c07quadraticepressions.indd 296 7/5/16 5:58 PM
ONLINE ONLY 7.7 int-2844 int-2845 int-3594 Review The Maths Quest Review is availale in a customisale format for students to demonstrate their knowledge of this topic. The Review contains: Fluency questions allowing students to demonstrate the skills they have developed to efficiently answer questions using the most appropriate methods Prolem Solving questions allowing students to demonstrate their aility to make smart choices, to model and investigate prolems, and to communicate solutions effectively. A summary of the key points covered and a concept map summary of this topic are availale as digital documents. Language inomial coeffi cient common factor completing the square difference of two squares epand Link to assesson for questions to test your readiness FOr learning, your progress as you learn and your levels OF achievement. Link assesson to assesson provides Voluptin sets of cupta questions quia volutem vit for harunti every topic se pario in your doassesson course, VoassessON well Voluptin as giving cupta instant quia feedack volutem and vit worked harunti se pario dolorporplorporro solutions to help improve quia dollitae your mathematical nia sim rempell orroreiumquo skills. core volum fugit voluptat. www.assesson.com.au epression factorise FOIL general form grouping three and one grouping two and two www.jacplus.com.au Review questions Download the Review questions document from the links found in your ebookplus. monic non-monic perfect squares quadratic trinomial term turning point form The story of mathematics is an eclusive Jacaranda video series that eplores the history of mathematics and how it helped shape the world we live in today. Adelard of Bath (eles-1846) delves into the world of a 12th-century English scholar whose translation of Araic tets into English allowed ideas to e spread that revolutionised the way mathematics was studied in Europe. Topic 7 Quadratic epressions 297
numer and algera <InVeSTIgaTIOn> InVeSTIgaTIOn FOr rich TaSK Or <numer and algera> FOr PuZZle rich TaSK 298 U 1 N C O R R EC TE D PA G E PR O O FS Celerity squares and doules Maths Quest 10 + 10A c07quadraticepressions.indd 298 7/5/16 5:50 PM
2 Beginning the game There is to e no communication etween players at this time. Your teacher will randomly allocate a headand to each player y placing a headand on their head without the player seeing the numer on their headand. 3 Playing the game The oject of the game The oject of the game is to use the process of elimination for you to fi nd your pair. A possile train of thought is illustrated at right. Starting the game Once all headands have een allocated, stand in a circle or walk around freely. Without speaking, determine who is a match; then, y a process of elimination, determine who might e your match. Hmm now, Lizzy is 16 and Jo is 8 so they are a match. Nick is 10. I can t see a 25 maye he is my match. Making a match When you think you have found your match, approach that person and say I think I am your match. The other player should now check to see if you have a match elsewhere and can reply y saying one of two things: Yes, I think I am your match, or I know your match is still out there. If a match is agreed upon, the players should sit out for the remainder of the game. If a match is not agreed upon, players should continue looking. Ending the game The class should continue until everyone is in a pair, at which time the class can check their results. The class should now discuss the different trains of thought they used to fi nd their pair and how this relates to factorising quadratic trinomials. 25 Topic 7 Quadratic epressions 299
<InVeSTIgaTIOn> numer and algera FOr rich TaSK Or <numer and algera> FOr PuZZle COde PuZZle Isaac Newton demonstrated this in 1665. Factorise the trinomials on the left of the page. Join the dot net to each with the dots net to its factors, using straight lines. The two lines will pass through a letter and numer giving the puzzle s code. 3 2 + 14 + 8 2 2 + 3 9 2 2 1 2 2 + 5 3 2 2 3 2 3 2 + 2 1 2 2 9 + 4 3 2 + 2 3 2 13 10 4 2 4 15 3 2 11 4 2 2 3 3 2 8 + 4 3 2 + 4 + 1 3 2 10 8 2 2 + 5 12 2 2 7 + 6 2 2 11 + 12 3 2 7 + 2 H T F 14 4 D S I 1 12 U 15 P 17 N O 9 3 C 16 W 7 X M R 19 G 10 5 2 A L 8 6 13 18 ( + 1) ( 1) ( 2) ( + 3) ( 4) ( + 4) ( 5) (2 + 1) (2 1) (2 + 3) (2 3) (2 5) (3 + 1) E 11 (3 1) (3 + 2) (3 2) 1 2 3 4 5 6 3 7 2 4 3 8 9 10 11 5 12 13 14 15 10 9 3 16 4 12 17 5 14 15 11 3 15 15 5 17 5 18 4 19 14 6 14 12 17 8 300 Maths Quest 10 + 10A
Activities 7.1 Overview Video The story of mathematics (eles-1846) 7.2 epanding algeraic epressions Interactivity IP interactivity 7.2 (int-4596): Epanding algeraic epressions digital docs SkillSHEET (doc-5244): Epanding rackets SkillSHEET (doc-5245): Epanding a pair of rackets 7.3 Factorising epressions with three terms Interactivity IP interactivity 7.3 (int-4597): Factorising epressions with three terms digital docs SkillSHEET (doc-5250): Finding a factor pair that adds to a given numer WorkSHEET 7.1 (doc-5251): Factorising and epanding 7.4 Factorising epressions with two or four terms Interactivity IP interactivity 7.4 (int-4598): Factorising epressions with two or four terms digital docs SkillSHEET (doc-5246): Factorising y taking out the highest common factor SkillSHEET (doc-5247): Factorising y taking out a common inomial factor To access eookplus activities, log on to 7.5 Factorising y completing the square Interactivities Completing the square (int-2783) IP interactivity 7.5 (int-4599): Factorising y completing the square 7.6 mied factorisation Interactivity IP interactivity 7.6 (int-4600): Mied factorisation digital docs SkillSHEET (doc-5248): Simplifying algeraic fractions SkillSHEET (doc-5249): Simplifying surds SkillSHEET (doc-5252): Factorising y grouping three and one WorkSHEET 7.2 (doc-5254): Mied factorisation 7.7 review Interactivities Word search (int-2844) Crossword (int-2845) Sudoku (int-3594) digital docs Topic summary (doc-13806) Concept map (doc-13807) www.jacplus.com.au Topic 7 Quadratic epressions 301
Answers topic 7 Quadratic epressions Eercise 7.2 Epanding algeraic epressions 1 a 2 + 6 4 20 c 21 3 d 3 e 2 + 2 f 2 2 8 g 15 2 6 h 10 15 2 i 8 2 + 2 j 4 3 6 2 k 6 3 3 2 l 15 3 + 20 2 2 a 2 12 2 2 3 c 2 5 14 d 2 6 + 5 e 2 + 6 f 2 6 + 8 g 2 2 17 + 21 h 3 2 2 i 6 2 17 + 5 j 21 17 + 2 2 k 15 + 14 8 2 l 110 + 47 21 2 3 a 2 2 4 6 8 2 28 16 c 2 2 + 12 + 14 d 2 3 2 e 3 3 75 f 6 3 54 g 2 3 12 2 + 18 h 5 3 30 2 + 40 i 6 3 6 2 + 120 4 a 3 + 2 2 2 3 2 2 5 + 6 c 3 5 2 + 5 d 3 6 2 + 11 6 e 2 3 7 2 5 + 4 f 6 3 7 2 + 1 5 a 2 2 2 2 + 4 + 10 c 5 2 6 5 d 19 23 e 5 1 f 2 + 6 g 2 2 3 +!3 h!6 + 2!2 3!3 6 2!5 6 a A C 7 B 8 a 2 2 + 1 2 + 4 + 4 c 2 + 10 + 25 d 16 + 8 + 2 e 49 14 + 2 f 144 24 + 2 g 9 2 6 + 1 h 144 2 72 + 9 i 25 2 + 20 + 4 j 4 12 + 9 2 k 25 40 + 16 2 l 1 10 + 25 2 9 a 2 2 12 + 18 4 2 56 + 196 c 3 2 + 6 + 3 d 4 2 12 9 e 49 2 + 14 1 f 8 2 24 + 18 g 12 + 108 243 2 h 45 + 330 605 2 i 16 2 16 4 10 a 2 49 2 81 c 2 25 d 2 1 e 4 2 9 f 9 2 1 g 49 2 h 64 2 i 9 4 2 11 a ( + 1)( 3) 2 2 3 c 6 cm, 2 cm, 12 cm 2 12 a m ( + 2) m ( + 1) m c ( + 1)( + 2) d 2 + 3 + 2 e 4 m 2, 12 m 2 13 a Student C Student B: (3 + 4)(2 + 5) 3 2 + 3 5 + 4 2 + 4 5 6 2 + 23 + 20 Student A: (3 + 4)(2 + 5) 3 2 + 3 5 + 4 2 + 4 5 6 2 + 15 + 8 + 20 6 2 + 23 + 20 14 (a )(a + ) a 2 2 LHS (5 3)(5 + 3) 2 8 16 RHS: 5 2 3 2 25 9 16 LHS RHS True 15 (a + ) 2 a 2 + 2a + 2 LHS: (5 + 3) 2 8 2 64 RHS: 5 2 + 2 5 3 + 3 2 25 + 30 + 9 64 LHS RHS True 16 Answers will vary; eamples are shown. a 1 + 4)( + 3) 2 + 7 + 12 1 + 4) 2 2 + 8 + 16 c 1 + 4)( 4) 2 16 d 1 + 4) 2 2 + 8 + 16 17 The square of a inomial is a trinomial; the difference of two squares has two terms. 18 (a + )(c + d) (c + d)(a + ) LHS: (a + )(c + d) ac + ad + c + d RHS: (c + d)(a + ) ca + c + da + d LHS RHS True 19 a 4 2 + 12y 20z + 9y 2 30yz + 25z 2 1 + 1 4 4 + 2 42 1 20 a V 6 3 29 2 + 46 24 TSA 12 2 + 20 + 3 Challenge 7.1 10, 11, 13, 18, 35 Eercise 7.3 Factorising epressions with three terms 1 a ( + 2)( + 1) ( + 3)( + 1) c ( + 8)( + 2) d ( + 4) 2 e ( 3)( + 1) f ( 4)( + 1) g ( 12)( + 1) h ( 6)( + 2) i ( + 4)( 1) j ( + 5)( 1) k ( + 7)( 1) l ( + 5)( 2) m ( 3)( 1) n ( 4)( 5) o ( + 14)( 5) 2 a 2( + 9)( + 1) 3( + 2)( + 1) c ( + 2)( + 1) d ( + 10)( + 1) e ( + 2)( + 5) f ( + 12)( + 1) g ( + 3)( + 4) h ( + 2)( + 6) i 2( + 2)( + 5) j 3( + 1)( + 10) k 5( + 20)( + 1) l 5( + 4)( + 5) 3 a (a 7)(a + 1) (t 4)(t 2) c ( + 4)( + 1) d (m + 5)(m 3) e (p 16)(p + 3) f (c + 16)(c 3) g (k + 19)(k + 3) h (s 19)(s + 3) 302 Maths Quest 10 + 10A
i (g + 8)(g 9) j (v 25)(v 3) k ( + 16)( 2) l ( 15)( 4) 4 a C B 5 C 6 a (2 + 1)( + 2) (2 1)( 1) c (4 + 3)( 5) d (2 1)(2 + 3) e ( 7)(2 + 5) f (3 + 1)( + 3) g (3 7)(2 1) h (4 7)(3 + 2) i (5 + 3)(2 3) j (4 1)(5 + 2) k (3 + 2)(4 1) l (3 1)(5 + 2) 7 a 2( 1)(2 + 3) 3(3 + 1)( 7) c 12(2 + 1)(3 1) d 3(3 + 1)(2 1) e 30(2 + 1)( 3) f 3a(4 7)(2 + 5) g 2(4 3)( 2) h (2 7)(5 + 2) i (8 1)(3 4) j 2(3 y)(2 + y) k 5(2 7y)(3 + 2y) l 12(5 + 3y)(10 + 7y) 8 a w 2 + 5w 6 (w + 6)(w 1) c ( + 5)( 2) 9 a ( + 5) ( + 5) c ( 1) 2 d ( + 9)( + 5) e ( 15)( 6) f ( 10)( 3) 10 ( 0.5)( + 1.5) 11 a ( 5)( + 1) ( 5) cm c 15 cm d 160 cm 2 e 3000( 5)( + 1) cm 2 or (3000 2 12 000 15 000) cm 2 12 a (2 + 3)(3 + 1) P 10 + 8 c 8 metres 13 a SA 3 2 + 16 Total area 3 2 + 16 + 16 c (3 + 4)( + 4) d l 21 m; w 7 m; d 2 m e 275 m 2 f 294 m 3 14 a Yellow 3 cm 3 cm Black 3 cm 6 cm White 6 cm 6 cm Yellow 0.36 m 2 Black 0.72 m 2 White 0.36 m 2 c 15 a 2 7 + 12 ( 3)( 4) 2 + 7 12 ( 3)( 4) c 2 + 2 ( 2)( + 1) d 2 4 21 ( + 3)( 7) e 2 + 4 21 ( 3)( + 7) f 2 30 ( + 5)( 6) g 2 + 7 8 ( 1)( + 8) h 2 11 + 30 ( 5)( 6) 16 a (9a 2)(6a 7) (3m 2 + 2)(m!7)(m +!7) c (2 sin() 1)(sin() 1) 17 a ( + 2!3)(2!3) 2z(z 2 + 3) Eercise 7.4 Factorising epressions with two or four terms 1 a ( + 3) ( 4) c 3( 2) d 4( + 4) e 3(3 1) f 8(1 ) g 3(4 ) h 4(2 3) i (8 11) 2 a ( 2)(3 + 2) ( + 3)(5 2) c ( 1)( + 5) d ( + 1)( 1) e ( + 4)( 2) f ( 3)(4 ) 3 a ( + 1)( 1) ( + 3)( 3) c ( + 5)( 5) d ( + 10)( 10) e (y + k)(y k) f (2 + 3y)(2 3y) g (4a + 7)(4a 7) h (5p + 6q)(5p 6q) i (1 + 10d)(1 10d) 4 a 4( + 1)( 1) 5( + 4)( 4) c a( + 3)( 3) d 2( + 2d )( 2d ) e 100( + 4)( 4) f 3a( + 7)( 7) g 4p( + 8)( 8) h 4(3 + 2)(3 2) i 3(6 + )(6 ) 5 a C B c B 6 C 7 a ( +!11)(!11) ( +!7)(!7) c ( +!15)(!15) d (2 +!13)(2!13) e (3 +!19)(3!19) f 3( +!22)(!22) g 5( +!3)(!3) h 2( +!2)(!2) i 12( +!3)(!3) 8 a ( 3)( + 1) ( 4)( + 6) c ( 5)( + 1) d ( 1)( + 7) e (6 )( + 8) f (10 )( + 2) g 8( 3) h (7 )(5 + 1) i ( 22)(9 + 2) 9 a ( 2y)(1 + a) ( + y)(2 + a) c ( y)(a + ) d ( + y)(4 + z) e (f 2)(e + 3) f (n 7)(m + 1) g 3(2r s)(t + u) h 7(m 3)(n + 5) i 2(8 j)(4 + k) j a(3 )(a + c) k (5 + y)( + 2) l m(m + n)(2 n) 10 a (y + 7)( 2) (m + 2)(n 3) c (q + 5)(p 3) d (s + 3)(s 4t) e ( + d)(a 2 c) f (1 + 5z)(y z) 11 a (a )(a + + 4) (p q)(p + q 3) c (m + n)(m n + l) d ( + y)(7 + y) e (1 2q)(5p + 1 + 2q) f (7g + 6h)(7g 6h 4) 12 a ( + 7 + y)( + 7 y) ( + 10 + y)( + 10 y) c (a 11 + )(a 11 ) d (3a + 2 + )(3a + 2 ) e (5p 4 + 3t)(5p 4 3t) f (6t 1 +!5v)(6t 1!5v) 13 a E A c D 14 B 15 a ( 5)( + 5) ( 5) cm, ( + 5) cm c 2 cm, 12 cm d 24 cm 2 e 120 cm 2 or 6 times igger 16 a A 1 πr 2 m 2 A 2 π(r + 1) 2 m 2 c A π(r + 1) 2 πr 2 π(2r + 1) m 2 d 34.56 m 2 17 a Annie ( + 3)( + 2) m 2 Bronwyn 5( + 2) m 2 ( + 3)( + 2) 5( + 2) c ( + 2)( 2) 2 4 d Width 5 m e Annie has 30 m 2 and Bronwyn has 25 m 2. 18 a ( 2y a + 3)( 2y + a 3) ( 1)( 2 + 10 + 37) 19 a (3 1)(9 2 + 3 + 1) 3(2 5y 3)(2 + 5y 3) Eercise 7.5 Factorising y completing the square 1 a 25 9 c 4 d 64 e 100 f 16 g 49 h 625 i 49 4 j 1 4 2 a ( 2 +!11)( 2!11) ( + 1 +!3)( + 1!3) c ( 5 +!13)( 5!13) d ( + 3 +!19)( + 3!19) e ( + 8 +!65)( + 8!65) f ( 7 +!6)( 7!6) g ( + 4 +!7)( + 4!7) h ( 2 +!17)( 2!17) i ( 6 +!11)( 6!11) 3 a a 1 2 +!5 2 a 1 2!5 2 a 3 2 +!21 2 a 3 2!21 2 Topic 7 Quadratic epressions 303
c a + 1 2 +!21 2 a + 1 2!21 2 d a + 3 2 +!13 2 a + 3 2!13 2 e a + 5 2 +!17 2 a + 5 2!17 2 f a + 5 2 +!33 2 a + 5 2!33 2 g a 7 +!53 a 7!53 2 2 2 2 h a 9 +!29 a 9!29 2 2 2 2 i a 1 2 +!13 2 a 1 2!13 2 4 a 2( + 1 +!3)( + 1!3) 4( 1 +!6)( 1!6) c 5( + 3 + 2!2)( + 3 2!2) d 3( 2 +!17)( 2!17) e 5( 3 +!7)( 3!7) f 6( + 2 +!5)( + 2!5) g 3( + 5 + 2!3)( + 5 2!3) h 2( 2 +!11)( 2!11) i 6( + 3 +!14)( + 3!14) 5 i d ii iii c iv a v c vi d vii d viii e 6 a B E 7 E 8 a 0.55; 5.45 9 Check with your teacher. 10 a 2a + 2!14!14 a + 2 + 2 2 This epression cannot e factorised as there is no difference of two squares. 11 ( p)( p + 2) Eercise 7.6 Mied factorisation 1 3( + 3) 2 ( + 2 + 3y)( + 2 3y) 3 ( + 6)( 6) 4 ( + 7)( 7) 5 (5 + 1)( 2) 6 5(3 4y) 7 (c + e)(5 + d) 8 5( + 4)( 4) 9 ( + 5)( + 1) 10 ( + 4)( 3) 11 (m + 1)(n + 1) 12 ( +!7)(!7) 13 4(4 1) 14 5( + 10)( + 2) 15 3(3 y)( + 2) 16 ( 4 + y)( 4 y) 17 4( 2 + 2) 18 (g + h)(f + 2) 19 ( +!5)(!5) 20 5(n + 1)(2m 1) 21 ( + 5)( + 1) 22 ( + 1)( 11) 23 ( + 2)( 2) 24 (a + )(c 5) 25 (y + 1)( 1) 26 (3 + 2)( + 1) 27 7( + 2)( 2) 28 4( + 6)( + 1) 29 (2 + r)(p s) 30 3( + 3)( 3) 31 (u + v)(t 3) 32 ( +!11)(!11) 33 (4 1)(3 1) 34 ( + 1)( 3) 35 ( + 6)( 2) 36 4( 1)( + 4) 37 3( + 2)( + 8) 38 (3 + )(7 ) 39 4(3 + 2y)(3 2y) 40 3(y + ) (y ) 41 4( + 2) 42 (3 4y)( 2y) 43 ( + 7)( + 4) 44 ( + 2)( 5) 45 (2 + 3)( + 3) 46 a ( + 5) ( 2) ( + 2) ( + 2) ( + 2) ( 2) ( 4) ( + 2) ( + 5) ( 2) ( + 2) ( + 2) ( + 2) ( 2) ( 4) ( + 2) c + 5 4 1 + 1 18 47 a c 6 2 + 3 ( 5) d 2 1 e + 2 6 f + 4 + 5 + 3 4( + 2) p(p + 7) 5(m + 2 + n) g h i 5 (p + 3)(p 2) 2(2m 5) 5(3d 2)(d 3 + 5e) j 4(d 2)(4d + 3) 48 2 16 9 25 49 a ( 9 y) ( 9 + y) (2 + 3 4y) (2 + 3 + 4y) 50 2 a 2 ; they are the same. a ( 13)( +13) or ( 13)( + 13) 36( 2c)( + 2c) or 36( 2c)( + 2c) c 2 y 2 (15 13y 2 )(15 +13y 2 ) or 2 y 2 ( 15 13y 2 )( 15 + 13y 2 ) 51 a ( + y) ( y + 3) ( + y) (7 + y) c (1 2q) (5p + 1 + 2q) 52 1 a 2 + 2a + 4 53 ( + 6 2y)( + 6 + 2y) Challenge 7.2 a 10609 3844 c 99409 d 1024144 e 2809 f 9604 Investigation Rich task Check with your teacher. Code puzzle White light is made up of a miture of colours. 304 Maths Quest 10 + 10A