Outside the high school program / Algebra - remarkable identity 38 exercises (100% corrected)

a
1. Develop E.8178 Expand the following expressions : a ( x + 2)( x − 2) b ( x + 1)( x − 1) c (2 x − 3)(2 x + 3) d (3 − 4 x )(3 + 4 x ) e (2 x + 2)(2 x − 2) f ( a + b )( a − b ) E.5340 Expand each of the following expres-sions : a (3 x + 2) 2 b (2 x − 5) 2 c (3 x + 8)(3 x − 8) d ( − 4 x − 1) 2 E.677 Using remarkable identities, deter-mine the expanded and reduced forms of the following ex-pressions : a (5 x + 6) 2 b (2 x − 6)(2 x + 6) c (8 − 4 x ) 2 d (2 x + 1)(2 x + 1) e (1 − x )(1 + x ) f (2 − x ) 2 E.2332 Complete the dotted lines below : a x − 3 2 = x 2 − : : : + 9 b 3 x + 1 2 = 9 x 2 + : : : + 1 c x − 2 2 = x 2 − : : : + 4 E.679 Copy and complete the following equalities so that they are true : a (2 x + : : : ) 2 = : : : + 20 x + : : : b ( : : : − : : : ) 2 = 81 x 2 − 36 x + : : : c ( : : : − 1)( : : : + 1) = 9 x 2 − : : : E.692 Copy and complete the dotted lines to verify the equality below : a ( x + : : : ) 2 = : : : + 6 x + : : : b ( : : : − : : : ) 2 = 4 x 2 − : : : + 25 c : : : − 64 = (7 x − : : : )( : : : + : : : ) 2. Factorize a remarkable identity E.9670 Factor the following expressions : a x 2 + 2 x + 1 b x 2 − 6 x + 9 c x 2 − 25 E.684 Factor the following expressions : a 9 x 2 − 42 x + 49 b 25 x 2 + 30 x + 9 c 9 x 2 − 4 E.9662 Factor the following expressions : a 81 x 2 − 126 x + 49 b 36 x 2 + 24 x + 4 c 9 x 2 + 12 x + 4 d 4 x 2 − 25 E.683 Factorize, if possible , the following literal expressions. If this is not possible, explain why. a 25 x 2 − 49 b 9 x 2 + 12 x − 4 c 100 x 2 + 50 x + 25 d 36 x 2 − 1 3. Factoring: a step further E.467 Each of the following expressions is factorable. Give the factorized form of each : a (2 x − 1) 2 − 4(2 − x ) 2 b (2 x + 1) 2 − 4(2 − 3 x ) 2 E.9673 Factor the following expressions : a 9 x 2 − 12 x + 4 + (4 − 3 x )(3 x − 2) b 25 x 2 − 49 − x (5 x + 7) E.9672 Factor the following expressions : a ( x − 2)(3 x − 2)+9 x 2 − 12 x +4 b (2 x + 1) 2 − (2 x − 1) 2 E.9677 Factor each of the following expres-sions : a ( x + 1) 2 − (2 x − 3) 2 b ( x + 1)(2 x − 3) − ( x 2 − 1) E.9674 Factor the following expressions : a 4 3 − 2 x 2 − 9 x − 3 2 b x 2 − 9(2 x − 1) 2 E.9679 Factor the following expressions : a 2 x ( x + 1) + ( x + 1)( x 2 + 1) b 12 x 2 − 6 x + (2 x − 1)(5 − 2 x ) https://chingmath.fr chapExoCorrec/8178 sacados/8178 chapExoCorrec/5340 sacados/5340 chapExoCorrec/677 sacados/677 chapExoCorrec/2332 sacados/2332 chapExoCorrec/679 sacados/679 chapExoCorrec/692 sacados/692 Groupement Est - Juin 2002 chapExoCorrec/9670 sacados/9670 chapExoCorrec/684 sacados/684 chapExoCorrec/9662 sacados/9662 chapExoCorrec/683 sacados/683 chapExoCorrec/467 sacados/467 chapExoCorrec/9673 sacados/9673 chapExoCorrec/9672 sacados/9672 chapExoCorrec/9677 sacados/9677 chapExoCorrec/9674 sacados/9674 chapExoCorrec/9679 sacados/9679
E.2857 Perform the following factorizations : a 3 x +3 2 − x +2 5 x +4 b 2 x − 4 4 x − 4 + x − 3 2 E.8508 Factorize the following factoriza-tions : a − 3 x − 1 x − 3 + 1 − 2 x 2 E.9668 Factor the following expressions : a (3 x + 1)(4 x + 5) + (3 x + 4)(5 − x ) b ( x + 2)(3 x + 2) − 2 x − 1 E.5901 Factor the following expressions : a ( x + 2) 2 + (3 x + 3)( x − 1) b ( x + 1)(3 x + 2) + (3 x − 1)(2 x + 1) c (2 x − 1) 2 − (3 x + 3)( x − 5) Hint: it is necessary to obtain the expanded-reduced form of each of its expressions to recognize a remarkable identity. 4. Square roots and remarkable identity E.9691 Expand and then simplify each of the expressions below : a 1 + 3 2 b 5 − 2 2 c 3 + 7 3 − 7 d 5 − 2 2 E.9692 Perform the following calculations using the remarkable identities and give the result in the sim-plest form possible : a 1 + 3 6 1 − 3 6 b 2 − 4 3 2 E.9693 Work out the calculations below and give their results in the form a + b c , where a , b , c are integers with c as small as possible : a 6 − 3 5 6 + 3 5 b 3 + 2 2 2 + 3 3 E.254 1 Write 2+ 3 2 as a + b 3 where a and b are real num-bers. 2 Deduce a writing simplification of 7+4 3 . 5. Expand, factor, and calculate E.676 Consider the expression : C =(2 x +5) 2 − ( x +3)(2 x +5) 1 Expand and reduce C . 2 Factorize C . 3 Calculate the expression C for x = − 2 3 . E.3757 Consider the expression : E =(2 x − 3) 2 +(2 x − 3)( x +8) 1 Expand then reduce the algebraic expression E . 2 Factor the algebraic expression E . 3 Calculate the expression E when x = 3 2 . E.687 1 Expand the expression : A =(2 x +4) 2 2 Give the factorized form of : B =4 x 2 +16 x +16 3 Give the value of B for x = − 2 . 6. Expansion, factoring and equations E.825 Using the method of your choice, solve the following equations : a 9 x 2 + 6 x + 1 = 0 b ( x + 1) 2 − (2 x − 1) 2 = 0 c x 2 + 2 x = − 1 E.5353 Consider the following two calcula-tion programs Program A : Choose a number; multiply it by 2 ; add 3 ; square it. Program B: Choose a number; multiply by 16 ; add 8 . 1 Give the output value of these two calculation programs when the starting value is 2 . 2 What number should be chosen so that both programs have the same output value? https://chingmath.fr chapExoCorrec/2857 sacados/2857 chapExoCorrec/8508 sacados/8508 chapExoCorrec/9668 sacados/9668 chapExoCorrec/5901 sacados/5901 chapExoCorrec/9691 sacados/9691 chapExoCorrec/9692 sacados/9692 chapExoCorrec/9693 sacados/9693 chapExoCorrec/254 sacados/254 chapExoCorrec/676 sacados/676 Groupe Est Juin 2003 5,5 points chapExoCorrec/3757 sacados/3757 chapExoCorrec/687 sacados/687 chapExoCorrec/825 sacados/825 chapExoCorrec/5353 sacados/5353
E.2508 1 Expand and reduce : A =(2 x − 1) 2 − 4(2 − x ) 2 Factor: B =( x − 1) 2 +(3 x +5)( x − 1) 3 Solve the equation : ( x − 1)(4 x +4)=0 E.2509 Consider the expression : A =( x − 3)( x +3) − 2( x − 3) 1 Factorize A . 2 Expand and reduce A . 3 Choosing the most suitable expression of A from those found in the previous questions, determine the value of A for x = − 1 and for x =0 . 4 Solve the equation : ( x − 3)( x +1)=0 E.834 Consider the expression : E =(3 x − 1)( x +5) − (3 x − 1) 2 1 Develop and reduce E 2 Factorize E . 3 Solve the equation : (3 x − 1)( − 2 x +6)=0 7. Equation reducing to 1st degree E.9681 Solve the equations : a ( x + 1) 2 − ( x − 1) 2 = 0 b 4 x 2 − 1 = (2 x + 2) 2 E.9683 Solve, by the method of your choice, the following equations : a x 2 + 2 x + 2 = ( x + 4) 2 b 3(2 x + 4) 2 = (6 x − 2)(2 x + 1) 8. Product equations: one step further E.477 Solve the following equations : a 2 x 2 + x + 1 = x 2 − x E.9682 Solve the equations : a ( x + 3)(2 x + 3) = x + 1 b (2 x − 2) 2 + ( x + 6)(5 x + 2) = 0 E.451 1 Expand the following two expressions : ( x − 1)( x + 2) 2 ; x 2 · ( x + 3) − 4 2 Deduce a method for solving the equation : x 2 ( x + 3) = 4 . 9. Unclassified financial years E.3755 Consider the algebraic expression : A = ( x − 3) 2 − x ( x − 4) 1 Determine the expanded and reduced form of A . 2 Find a value of x for which A =9 . https://chingmath.fr chapExoCorrec/2508 sacados/2508 Asie du Sud-Est - Juin 2003 chapExoCorrec/2509 sacados/2509 chapExoCorrec/834 sacados/834 chapExoCorrec/9681 sacados/9681 chapExoCorrec/9683 sacados/9683 chapExoCorrec/477 sacados/477 chapExoCorrec/9682 sacados/9682 chapExoCorrec/451 sacados/451 chapExoCorrec/3755 sacados/3755