Grade 9 / Remarkable identities 48 exercises (100% corrected)

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123420km6km6km20km abFig.1abFig.2abFig.3abFig.4 abaa−bb ChingQuizz : 8 exercises available for Quizz assessment : 1. First approach E.9208 1 Consider the square shown below that has been divided into four parts (two rectangles and two squares) : a Determine the area of the four parts of this square. b Derive, without using the calculator, the value of 26 2 . 2 Use the same method to calculate the value of 107 2 . E.9209 Let a and b be two strictly positive real numbers. Consider the four representations of the same square of side a below : 1 a Express using the numbers a and b the area of each of the hatched parts. b Which part of this figure admits the area of the expres-sion : a − b 2 +2 ab − b 2 2 Justify identity: a − b 2 = a 2 − 2 ab + b 2 E.7998 Consider the two figures below. One is shaded and the other is composed of two hatched figures : Show that these two figures have the same area 2. Introduction to development E.701 Expand and reduce the following ex-pressions : a ( x + 1)( x + 1) b ( x + 6)( x + 6) E.694 Entwickeln und reduzieren Sie die folgenden Ausdrücke : a ( x − 2)( x − 2) b ( x − 3)( x − 3) E.2219 Expand and reduce the following expressions : a ( x + 2)( x − 2) b ( x + 1)( x − 1) E.690 Expand and reduce the following lit-eral expressions : a (4 x − 7)(4 x + 7) b (3 x − 2)(3 x + 2) E.11343 Expand and reduce the following expressions : a (2 x + 3)(2 x + 3) b (5 x + 1)(5 x + 1) E.11345 Expand and reduce the following expressions : a (2 x − 3)(2 x + 3) b (3 − 4 x )(3 + 4 x ) E.11344 Expand and reduce the following expressions : a (3 x − 1)(3 x − 1) b (5 x − 1)(5 x − 1) E.10737 Expand and reduce the following expressions : a (3 x + 3)(3 x + 3) b ( a + b ) 2 https://chingmath.fr chapExoCorrec/9208 sacados/9208 123420km6km6km20km chapExoCorrec/9209 sacados/9209 abFig.1abFig.2abFig.3abFig.4 chapExoCorrec/7998 sacados/7998 abaa−bb chapExoCorrec/701 sacados/701 chapExoCorrec/694 sacados/694 chapExoCorrec/2219 sacados/2219 chapExoCorrec/690 sacados/690 chapExoCorrec/11343 sacados/11343 chapExoCorrec/11345 sacados/11345 chapExoCorrec/11344 sacados/11344 chapExoCorrec/10737 sacados/10737
E.10738 Expand and reduce the following expressions : a (3 x − 2)(3 x − 2) b ( a − b ) 2 E.10739 Expand and reduce the following expressions : a (2 x + 2)(2 x − 2) b ( a + b )( a − b ) 3. Introduction to remarkable identities - development E.9391 Proposition: for all numbers a and b , we have the follow-ing identities: a + b 2 = a 2 + 2 × a × b + b 2 a − b 2 = a 2 − 2 × a × b + b 2 a + b a − b = a 2 − b 2 Expand and reduce the following expressions : a 2 x + 1 2 b 3 x − 4 2 c x − 2 x + 2 E.9390 By expanding and reducing the ex-pression on the left, establish the identities below : a x + 5 2 = x 2 +10 x +25 b x − 2 2 = x 2 − 4 x + 4 c x +5 x − 5 = x 2 − 25 E.9206 Expand and reduce the following literal expressions : a (3 x − 5) 2 b (4 x + 3) 2 c (3 x + 2) 2 d (2 − 5 x ) 2 E.3698 Give the developed form and reduce the following expressions : a (3 x + 1)(3 x − 1) b (2 x − 1) 2 c ( x + 3) 2 E.10744 Expand and reduce the following literal expressions : a 2 x + 1 2 b 3 x − 4 2 c 7 x + 3 7 x − 3 E.10745 Expand and reduce the following literal expressions : a 5 x − 4 2 b 3 x + 7 2 c x − 8 x + 8 E.10746 Expand and reduce the following literal expressions : a 2 − x x + 2 b 3 + 2 x 2 c 9 x − 5 2 E.10812 Give the expanded and reduced forms of the following expressions : a 5 x − 4 5 x + 4 b 3 x + 7 2 c 2 x − 1 2 E.10789 Expand and reduce the following expressions : a 1 3 x − 6 2 b 2 x − 1 4 2 c 1 3 x + 1 x − 3 E.9392 Expand and reduce the following expressions : a 1 4 x + 2 3 2 b 5 3 x − 1 3 2 c 5 4 x + 1 5 5 4 x − 1 5 4. Development E.698 Expand and reduce the following ex-pressions : a (4 x − 2) 2 − 2( x + 2) b (3 x + 2) 2 + 2( x + 4) E.10813 Expand and simplify the following expressions : a 2 x − 3 2 + 5 x +6 2 b 3+2 x 3 − 2 x + 5 x − 2 2 E.9205 Expand and reduce the following expressions : a (2 x + 1)(2 x − 1) + 4 × 2 + 3( x + 1) b (5 x + 1)(5 x − 1) − ( x + 1)(5 − x ) 5. Use of the development of remarkable identities E.3895 The following calculation program is considered : Choose a starting number Multiply this number by ( − 2) Add 5 to product Multiply the result by 5 Write the result obtained. https://chingmath.fr chapExoCorrec/10738 sacados/10738 chapExoCorrec/10739 sacados/10739 chapExoCorrec/9391 sacados/9391 chapExoCorrec/9390 sacados/9390 chapExoCorrec/9206 sacados/9206 chapExoCorrec/3698 sacados/3698 chapExoCorrec/10744 sacados/10744 chapExoCorrec/10745 sacados/10745 chapExoCorrec/10746 sacados/10746 chapExoCorrec/10812 sacados/10812 chapExoCorrec/10789 sacados/10789 chapExoCorrec/9392 sacados/9392 chapExoCorrec/698 sacados/698 chapExoCorrec/10813 sacados/10813 chapExoCorrec/9205 sacados/9205 chapExoCorrec/3895 sacados/3895
1 a Verify that when the starting number is 2 , we get 5 . b When the starting number is 3 , what result do we get? 2 What number must be chosen at the start for the result to be 0 ? 3 Arthur claims that, for any starting number x , the ex-pression ( x − 5) 2 − x 2 yields the result of the calculation program. Is he right? E.5240 We consider the following calcula-tion programs : Program A Choose a number Add to it 1 Calculate the square of the sum onbtained Subtract from the result the square of the starting number Program B Choose a number Add 1 to double that number 1 We choose 5 as the starting number. What result do we get with each of the two programs? 2 Show that no matter what number is chosen, the results obtained with the two programs are always equal. E.5658 The following calculation program is considered : Choose a number Add 2 Square this sum Subtract from the result the square of the number origi-nally chosen Subtract from the result 4 . 1 a Show that, if the number chosen is 1 , the calculation program returns the number 4 . b Show that, if the number chosen is 5 , the calculation program returns the number 20 . 2 What is the number returned by the calculation program if the number chosen is 2 ? 3 a Noting x as the starting number, what is the literal expression obtained by this calculation program? b Justify that the expression obtained by the calculation program is equal to 4 × x . E.3696 1 Expand : ( x − 1) 2 . Justify that 99 2 =9 801 using the previous expansion. 2 Expand : ( x − 1)( x +1) . Justify that 99 × 101=9 999 using the previous expanded form E.5672 Tom has to calculate 3 ; 5 2 . ˇ No need to use a calculator ı, says Julie, just multiply 3 by 4 and add 0 ; 25 . 1 Perform the calculation suggested by Julie and check that the result is indeed the square of 3 ; 5 . 2 Suggest a simple way to calculate 7 ; 5 2 and give the result. 3 Julie proposes the following conjecture : ( n + 0 ; 5) 2 = n ( n + 1) + 0 ; 25 where n is a positive integer. Prove that Julie’s conjecture is true (for any number n ) . 6. Factorisation and remarkable identity E.7997 Factor the following expressions : a x + 4 2 − 2 2 b x + 1 2 − 3 2 c x − 2 2 − 2 2 d 4 2 − x + 1 2 E.9207 Factor the following expressions, no explanations are required : B = ( x + 2) 2 − (1 − 2 x ) 2 ; C = 4 − 16 x 2 + (4 x + 2)(5 x − 4) 7. Expand, factor, and calculate E.699 1 Expand the expression : A =(2 x − 1) 2 . 2 Find the factored form of : B =4 x 2 − 4 x +1 3 Find the value of B when x =0 and when x = 1 2 . E.696 Consider the literal expression : ( E ) : (2 x − 1) 2 − 3( x + 1)(2 x − 1) 1 Factorize ( E ) . 2 Develop ( E ) . 3 Calculate ( E ) for x =1 in three different ways. Then calculate ( E ) for x = 1 4 . Subsidiary question: 4 What are the two values of x that cancel the expression https://chingmath.fr chapExoCorrec/5240 sacados/5240 chapExoCorrec/5658 sacados/5658 chapExoCorrec/3696 sacados/3696 chapExoCorrec/5672 sacados/5672 chapExoCorrec/7997 sacados/7997 chapExoCorrec/9207 sacados/9207 chapExoCorrec/699 sacados/699 chapExoCorrec/696 sacados/696
123456789101112ABCDEx0123456789Entierimpair2x135791113151719Entierimpairsuivant2x3579111315171921Produitdecesentiersimpairsconsécutifs(2xx315356399143195255323399Résultatobtenu(2xx4163664100144196256324400 Motif 1Motif 2Motif 3 ( E ) ? Can you justify? E.3697 Consider the expression : D = (2 x + 3) 2 + ( x − 5)(2 x + 3) 1 Expand and reduce the expression D . 2 Factor the expression D . 3 Evaluate the expression for x =1 and x = 2 3 . E.3695 Consider the expression : A = 1 4 ( a + b ) 2 − ( a − b ) 2 1 Calculate A for a =1 and b =5 . 2 Calculate A for a = − 2 and b = − 3 . 3 Alex states that the number A is equal to the product of the numbers a and b . Is he correct? Justify. 8. Problems E.6303 Lea thinks that by multiply-ing two consecutive odd integers (i.e., that follow each other) and adding 1 , the result is always a multiple of 4 1 Study an example: 5 and 7 are two consecutive odd integers. a Calculate: 5 × 7+1 . b Is Lea right about this example? 2 The table below shows the work she did in a spreadsheet. a From this table, what result do we get by taking as the first odd integer 17 ? b Show that this integer is a multiple of 4 . c Of the following four spreadsheet formulas, two for-mulas could be entered in cell D3 . Which ones? No justification is expected. Formula 1 : =(2*A3+1)*(2*A3+3) Formula 2 : =(2*B3+1)*(2*C3+3) Formula 3 : =B3*C3 Formula 4 : =(2*D3+1)*(2*D3+3) 3 Algebraic study: a Expand and reduce the expression : (2 x + 1)(2 x + 3) + 1 b Show that Lea was right : the result obtained is always a multiple of 4 . E.7634 Gaspard creates patterns with white and grey mosaic tiles in the following way: Gaspard forms a square with gray tiles and then borders it with white tiles. 1 How many white tiles will Gaspard use to border the gray square in the pattern 4 (a square with 4 gray tiles on each side) ? 2 a Justify that Gaspard can make a pattern like this using exactly 144 gray tiles. b How many white tiles will he then use to border the resulting gray square? 3 The pattern for which a square of n gray tiles is bordered is called ˇ pattern n ı. Three students each came up with an expression to calcu-late the number of white tiles needed to make the ˇ pat-tern n ı : Expression n o 1: 2 × n +2 × n +2 Expression n o 2: 4 × n +2 Expression n o 3: 4 × n +2 − 4 Only one of these three expressions is not appropriate. Which one? E.5183 For this exercise, any trace of re-search, even if incomplete, will be considered in the assess-ment. The following calculation program is given : Choose a number. Add to it 1 . Calculate the square of this sum. Remove 16 from the result. and the literal expression P defined by: P = x 2 +2 x − 15 Show that no matter what starting number is chosen, we get the same value : by the calculation program ; than by evaluating the expression P into this value. https://chingmath.fr chapExoCorrec/3697 sacados/3697 Antilles-Guyane Septembre 2008 chapExoCorrec/3695 sacados/3695 Liban Juin 2009 chapExoCorrec/6303 sacados/6303 123456789101112ABCDEx0123456789Entierimpair2x135791113151719Entierimpairsuivant2x3579111315171921Produitdecesentiersimpairsconsécutifs(2xx315356399143195255323399Résultatobtenu(2xx4163664100144196256324400 chapExoCorrec/7634 sacados/7634 Asie Juin 2017 7 points Motif 1Motif 2Motif 3 chapExoCorrec/5183 sacados/5183
ABCDEFGHIJ2x4 ABCDEFG 1cm4cmABCDMNP ABC4x3x5x ABC10x−56x−38x−4 9. Problems and geometry E.826 In the figure below AEFG , AHIJ and ABCD are squares. Let x be the measure of segment [ HB ] . 1 a Express AH as a function of x . b deduce the area of AHIJ . c Which of the algebraic expressions (or which) corre-spond to the hatched area: M =(4 − x ) 2 − 2 2 ; N =(4 − x − 2) 2 ; P =4 2 − x 2 − 2 2 2 Expand and reduce the expression : Q =(4 − x ) 2 − 4 . 3 Factorize Q . 4 Calculate Q for x =2 . What does this result mean for the figure? E.5917 The drawing below represents a compound figure : of a square ABCD and a rectangle DEFG . E is a point on segment [ AD ] . C is a point on segment [ DG ] . In this figure, the length AB can vary but we always have : AE = 15 cm ; CG = 25 cm 1 In this question, we assume that : AB =40 cm a Calculate the area of the square ABCD . b Calculate the area of the rectangle DEFG . 2 Can we find the length AB so that the area of the square ABCD is equal to the area of the rectangle DEFG ? If yes, calculate AB . If no, explain why. If the work is not completed, still leave a record of the search. It will be taken into account in the scoring . E.10814 Shown below is a square ABCD and a rectangle AMNP constructed from the sides of the square : We have : DP =1 cm ; BM = 4 cm Let x be the measure of the sides of the square ABCD . De-termine the value of x so that the square ABCD and the rectangle AMNP have the same area. Indication : all traces of research and explanations of the process will be taken into account in the assessment. E.680 Show that the triangle ABC is right-angled at A whatever the value of ˇ x ı : E.2601 Show that the triangle ABC is right-angled at C regardless of the value of ˇ x ı : 10. Share E.693 1 Expand and simplify the following expressions : a (3 x + 2) 2 b (5 − x )(5 + x ) + ( x − 1) 2 2 Factor the following expressions : a 9 x 2 − 25 b ( x + 1)(5 − 2 x ) − ( x + 1) 2 https://chingmath.fr chapExoCorrec/826 sacados/826 ABCDEFGHIJ2x4 chapExoCorrec/5917 sacados/5917 ABCDEFG chapExoCorrec/10814 sacados/10814 1cm4cmABCDMNP chapExoCorrec/680 sacados/680 ABC4x3x5x chapExoCorrec/2601 sacados/2601 ABC10x−56x−38x−4 chapExoCorrec/693 sacados/693
ABCx75x8 E.3759 This exercise is a multiple choice (MCQ) . No justification is required. For each of the questions, three answers are given, only one of which is correct. Each answer gives one point, a wrong answer or no answer takes away no points. For each question, indicate on the copy the number of the question and copy the correct answer. Réponse 1 Réponse 2 Réponse 3 1 6 − 4( x − 2) is equal to 2 x − 4 14 − 4 x − 2 − 4 x 2 What is the factor-ized expression of : 4 x 2 − 12 x +9 (2 x +3)(2 x − 3) (2 x + 3) 2 (2 x − 3) 2 3 For x = − 2 , the ex-pression 5 x 2 +2 x − 3 is equal to à 13 − 27 17 11. Unclassified exercises E.819 x is a positive number between 0 and 10 ; lengths are expressed in cm and areas in cm 2 . The figure below is made freehand. The question is whether there is a value of x for which ABC is a right-angled triangle. 1 Calculate AB and AC when x =4 . When x =4 , is ABC a rectangle? Justify answer. 2 Expand and reduce : ( x +7) 2 ; ( x +8) 2 Deduct : AB 2 − AC 2 =2 x +15 What is the value of AB 2 − AC 2 when x =0 , when x =10 ? Does the value of BC 2 depend on the number x ? 3 a Solve equation : 2 x +15=25 . b Deduce the value of x , so that the triangle ABC is right-angled at C . Justify your result. https://chingmath.fr chapExoCorrec/3759 sacados/3759 chapExoCorrec/819 sacados/819 ABCx75x8