WISKUNDE
GRAAD 10
NOG OEFENINGE
  
Gemengde voorbeelde : antwoorde.
  
MATHEMATICS
GRADE 10
MORE EXERCISES
  
Mixed examples : answers.
  
  
  
                               a4 ─ b4
        1.   ─────────────────
                 (a2 + 2ab + b2)(a2 + b2)
  
                             (a2 ─ b2)(a2 + b2)
                      =  ─────────────
                             (a + b)2(a2 + b2)
  
                             (a ─ b)(a + b)(a2 + b2)
                      =  ────────────────
                                 (a + b)2(a2 + b2)
  
                            a ─ b
                      =  ─────
                            a + b
  
                                                                           Vr. / Qu. 1.
  
  
  
                 a ─ 3b            3a ─ b
        3.   ──────  ÷  ───────
                2a + b             4a + 2b
  
                             a ─ 3b         2(2a + b)
                      =  ────── X ────────
                            2a + b            3a ─ b
  
                            2(a ─ 3b)
                      =  ──────
                             3a ─ b
  
                                                                           Vr. / Qu. 3.
  
  
  
  
  
                 2x2 + 3x ─ 9            2x + 6
        5.   ──────────  ÷  ───────
                2x2 + 5x ─ 12          2x2 + 8x
  
                         (2x ─ 3)(x + 3)           2x(x + 4)
                  =  ─────────── X ────────
                         (2x ─ 3)(x + 4)           2(x + 3)
  
                         2x
                  =  ───    =  x
                          2
  
                                                                           Vr. / Qu. 5.
  
  
  
  
                6x2 ─ 5x ─ 6            9x2 ─ 4
        7.   ──────────  ÷  ────────
                    4x2 ─ 9               2x2 + x ─ 3
  
                        (2x ─ 3)(3x + 2)         (2x + 3)(x ─ 1)
                  =  ───────────  X  ───────────
                        (2x ─ 3)(2x + 3)         (3x ─ 2)(3x + 2)
  
                          x ─ 1
                  =  ──────
                        3x ─ 2
  
                                                                           Vr. / Qu. 7.
  
  
  
  
                      c2 ─ 16              2c + 8
        9.   ──────────  ÷  ──────
                   c2 ─ 8c + 16           3c ─ 9
  
                        (c ─ 4)(c + 4)             3(c ─ 3)
                  =  ───────────  X  ──────
                             (c ─ 4)2                 2(c + 4)
  
                        3(c ─ 3)
                  =  ──────
                        2(c ─ 4)
  
                                                                           Vr. / Qu. 9.
  
  
                          3                          2x                  3
      11.   ──────────  ┼  ──────  ━  ──────
                9x2 ─ 12x + 4          9x2 ─ 4          3x ─ 2
  
                            3                           2x                     3
                =  ───────  ┼  ────────── ─ ─────
                     (3x ─ 2)2          (3x ─ 2)(3x + 2)      3x ─ 2
  
  
                     3(3x + 2) + 2x(3x ─ 2) ─ 3(3x ─ 2)(3x + 2)
               =  ───────────────────────────
                                        (3x ─ 2)2(3x + 2)
  
  
                     9x + 6 + 6x2 ─ 4x ─ 27x2 + 12
               =  ────────────────────
                                 (3x ─ 2)2(3x + 2)
  
                       ─21x2 + 5x + 18
               =  ──────────────
                        (3x ─ 2)2(3x + 2)
  
                                                                      Vr. / Qu. 11.
  
  
  
                a + x                a ─ x               2ax
      13.   ──────  ┼  ──────  ━  ──────
               2a ─ 2x           2a + 2x         a2 ─ x2
  
                         a + x              a ─ x                2ax
                =  ──────  ┼  ────── ─ ─────────
                     2(a ─ x)          2(a + x)       (a ─ x)(a + x)
  
  
                     (a + x)2 + (a ─ x)2 ─ 2(2ax)
               =  ───────────────────
                               2(a ─ x)(a + x)
  
  
                     a2 + 2ax + x2 + a2 ─ 2ax + x2 ─ 4ax
               =  ────────────────────────
                                      2(a ─ x)(a + x)
  
                       2a2 ─ 4ax + 2x2
               =  ──────────────
                        2(a ─ x)(a + x)
  
                       2(a2 ─ 2ax + x2)
               =  ──────────────
                        2(a ─ x)(a + x)
  
                          2(a ─ x)2
               =  ───────────
                     2(a ─ x)(a + x)
  
                     a ─ x
               =  ─────
                     a + x
  
                                                                      Vr. / Qu. 13.
  
  
  
  
                 a + b            4a2 ─ 9b2
        2.   ──────  X  ───────
               2a ─ 3b          a2 ─ b2
  
                             a + b          (2a ─ 3b)(2a + 3b)
                      =  ────── X ────────────
                           2a ─ 3b            (a ─ b)(a + b)
  
                            2a + 3b
                      =  ──────
                              a ─ b
  
                                                                           Vr. / Qu. 2.
  
  
  
  
  
  
  
  
  
                 a ─ b             a2 + 2ab + b2
        4.   ──────  X  ──────────
                 a + b              a2 ─ 2ab + b2
  
                             a ─ b           (a + b)2
                      =  ────── X ───────
                             a + b            (a ─ b)2
  
                            a + b
                      =  ─────
                            a ─ b
  
                                                                           Vr. / Qu. 4.
  
  
  
                 6x2 ─ 5x ─ 6                9x2 ─ 4
        6.   ──────────  ÷  ──────────
                      4x2 ─ 9              2x2 + x ─ 3
  
                       (2x ─ 3)(3x + 2)          (x ─ 1)(2x + 3)
                  =  ─────────── X ───────────
                       (2x ─ 3)(2x + 3)         (3x ─ 2)(3x + 2)
  
                         x ─ 1
                  =  ─────
                        3x ─ 2
  
                                                                           Vr. / Qu. 6.
  
  
  
                2x2 + xy ─ 6y2           9x2 ─ 4y2
        8.   ───────────  X  ───────
               3x2 + 4xy ─ 4y2          4x2 ─ 9y2
  
                       (x + 2y)(2x ─ 3y)         (3x ─ 2y)(3x + 2y)
                  =  ───────────  X  ─────────────
                       (x + 2y)(3x ─ 2y)         (2x ─ 3y)(2x + 3y)
  
                         3x + 2y
                  =  ─────
                        2x + 3y
  
                                                                           Vr. / Qu. 8.
  
  
  
  
  
                  3x2 ─ 7x + 2              x2 ─ 9
      10.     ─────────  X  ─────────
                  2x2 ─ 5x ─ 3        9x2 ─ 6x + 1
  
                        (x ─ 2)(3x ─ 1)           (x ─ 3)(x + 3)
                  =  ───────────  X  ───────────
                        (2x + 1)(x ─ 3)           (3x ─ 1)(3x ─ 1)
  
                         3x + 2y
                  =  ─────
                        2x + 3y
  
                                                                           Vr. / Qu. 10.
  
  
  
                         5                        3a                  3
     12.   ──────────  ┼  ──────  ━  ──────
               4a2 ─ 12a + 9          4a2 ─ 9          2a ─ 3
  
                            5                           3a                     3
                =  ───────  ┼  ────────── ─ ─────
                     (2a ─ 3)2         (2a ─ 3)(2a + 3)      2a ─ 3
  
  
                     5(2a + 3) + 3a(2a ─ 3) ─ 3(2a ─ 3)(2a + 3)
               =  ───────────────────────────
                                        (2a ─ 3)2(2a + 3)
  
  
                     10a + 15 + 6a2 ─ 9a ─ 12a2 + 27
               =  ───────────────────────
                                 (2a ─ 3)2(2a + 3)
  
                       ─6a2 + a + 42
               =  ──────────────
                        (2a ─ 3)2(2a + 3)
  
                                                                      Vr. / Qu. 12.
  
  
  
                  3y + 6                    12
     14.   ────────  ┼  ────────
               y2 ─ y ─ 6           y2 ─ y ─ 6
  
                              3y + 6                          12
                  =  ───────────  ┼  ─────────
                         (y + 2)(y ─ 3)           (y + 2)(y ─ 3)
  
                             3y + 18
                  =  ──────────
                        (y + 2)(y ─ 3)
  
                                                                           Vr. / Qu. 14.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
                3x + 2                  x + 5                      3x
      15.   ──────  ┼  ────────  ━  ────────
                x2 ─ 1           2x2 + x ─ 1        2x2 ─ 3x + 1
  
  
                         3x + 2                          x + 5                        3x
                =  ─────────  ┼  ────────── ─ ───────────
                     (x ─ 1)(x + 1)         (2x ─ 1)(x + 1)       (2x ─ 1)(x ─ 1)
  
  
                     (3x + 2)(2x ─ 1) + (x + 5)(x ─ 1) ─ 3x(x + 1)
               =  ─────────────────────────────
                                          (x ─ 1)(x + 1)(2x ─ 1)
  
  
                     6x2 + x ─ 2 + x2 + 4x ─ 5 ─ 3x2 ─ 3x
               =  ─────────────────────────
                                    (x ─ 1)(x + 1)(2x ─ 1)
  
                               4x2 + 2x ─ 7
               =  ────────────────
                       (x ─ 1)(x + 1)(2x ─ 1)
  
                                                                      Vr. / Qu. 15.
  
  
  
                      c + 2d                             c + d
     16.   ───────────  ━  ─────────────
               c2 + 4cd + 3d2           4c2 + 20cd + 24d2
  
  
                              c + 2d                              c + d
                  =  ───────────  ━  ────────────
                         (c + d)(c + 3d)          4(c + 2d)(c + 3d)
  
  
                             (c + 2d) X 4(c + 2d)  ─  (c + d) X (c + d)
                  =  ─────────────────────────────
                                       4(c + d)(c + 2d)(c + 3d)
  
  
                             4c2 + 16cd + 16d2  ─  c2 ─ 2cd ─ d2
                  =  ─────────────────────────────
                                       4(c + d)(c + 2d)(c + 3d)
  
  
                             3c2 + 14cd + 15d2
                  =  ──────────────────
                         4(c + d)(c + 2d)(c + 3d)
  
  
                             (c + 3d)(3c + 5d)
                  =  ──────────────────
                        4(c + d)(c + 2d)(c + 3d)
  
  
                               3c + 5d
                  =  ────────────
                        4(c + d)(c + 2d)
  
                                                                           Vr. / Qu. 16.
  
  
  
                   a2 ─ 2a                 3a                     5a
      17.   ────────  ┼  ──────  ━  ──────────
                a2 ─ a ─ 2           6a ─ 4           6a2 + 2a ─ 4
  
                      a(a ─ 2)                         3a                      5a
                =  ──────────  ┼  ─────── ─ ────────────
                     (a ─ 2)(a + 1)           2(3a ─ 2)         2(a + 1)(3a ─ 2)
  
  
                     a X 2(a ─ 2)(3a ─ 2) + 3a(a ─ 2)(a + 1) ─ 5a(a ─ 2)
               =  ──────────────────────────────────
                                            2(a ─ 2)(a + 1)(3a ─ 2)
  
  
                     6a3 ─ 16a2 + 8a + 3a3 ─ 3a2 ─ 6a ─ 5a2 + 10a
               =  ────────────────────────────────
                                      2(a ─ 2)(a + 1)(3a ─ 2)
  
                          9a3 ─ 24a2 + 12a
               =  ────────────────
                     2(a ─ 2)(a + 1)(3a ─ 2)
  
                           3a(a ─ 2)(3a ─ 2)
               =  ────────────────
                     2(a ─ 2)(a + 1)(3a ─ 2)
  
                          3a
               =  ───────
                     2(a + 1)                                                                       Vr. / Qu. 17.
  
  
  
  
                   3                     1                   a ─ 2x
     18.   ──────  ━  ───────  ━  ────────
              8a ─ 8x           8(a + x)           4(a2 + x2)
  
  
                         3(a + x)(a2 + x2)  ─  1(a ─ x)(a2 + x2)  ─  (a ─ 2x) X 2(a ─ x)(a + x)
                  =  ────────────────────────────────────────────
                                           8(a ─ x)(a + x)(a2 + x2)
  
  
                         3a3 + 3ax2 + 3a2x + 3x3 ─ a3 ─ ax2 + a2x + x3 ─ 2a3 + 2ax2 + 4a2x ─ 4x3
                  =  ─────────────────────────────────────────────────
                                                                 8(a ─ x)(a + x)(a2 + x2)
  
  
                               4ax2 + 8a2x
                  =  ─────────────────
                         8(a ─ x)(a + x)(a2 + x2)
  
  
                               4ax(x + 2a)
                  =  ─────────────────
                         8(a ─ x)(a + x)(a2 + x2)
  
  
                               ax(2a + x)
                  =  ─────────────────
                         2(a ─ x)(a + x)(a2 + x2)
  
  
                                                                           Vr. / Qu. 18.
  
  
  
  
'n Paar notas.  /  Some notes.
  
  
      Om al die terme in dalende, of stygende, magte
  
      van die onbekende te rangskik, is dit soms nodig
  
      om ook sommige terme se tekens te verander.
  
      Doen dit deur die terme in 'n negatiewe hakie
  
      te plaas en AL die tekens binnein die hakie
  
      te verander. Onthou dat as die teller en die
  
      noemer dieselfde teken het, is die breuk positief
  
      en verander die teken voor die breuk nie.
  
      As hulle verskillende tekens het moet
  
      die teken voor die breuk ook verander word.
  
  
  
  
      To arrange al the terms in descending, or ascending,
  
      order of the unknown, it is sometimes necessary
  
      to change the sign of some terms.
  
      Do this by placing the terms in a negative bracket
  
      and change the signs of ALL the terms in the bracket.
  
      Remember that if the numerator and the denominator
  
      have the same sign, the fraction is positive and
  
      the sign in front of the fraction does not change.
  
      If they have different signs, the fraction is
  
      negative and the sign in front of the fraction
  
      must be changed.
  
  
      By voorbeeld / For example :
  
                                             1 ─ a             ─(─1 + a)           verander die tekens in die negatiewe hakie
                                       ─  ─────   =   ───────
                                             3 ─ b             ─(─3 + b)           change the signs inside the negative bracket.
  
  
                                                                                              Herrangskik. Teken voor breuk bly dieselfde - teller
                                                                      a ─ 1               en noemer beide negatief en breuk dus positief.
                                                            =  ─  ──────
                                                                      b ─ 3               Rearrange. Sign in front of fraction remains the same -
                                                                                             numerator and denominator both negative and
                                                                                             fraction thus positive.
  
                                                                                            
  
  
  
  
                                                                                             verander die tekens in die negatiewe hakie
                                             1 ─ a             ─(─1 + a)           - teller; noemer positief geen verandering
                                       ─  ─────   =  ━ ───────
                                             3 + b                 3 + b               change the signs inside the negative bracket -
                                                                                             numerator; denominator positive, do not change.
  
  
                                                                                             Herrangskik. Teken voor breuk verander -
                                                                                             teller negatief en noemer positief breuk
                                                                      a ─ 1               dus negatief en teken voor breuk verander
                                                           =  ╋  ──────
                                                                      b + 3               Rearrange. Sign in front of fraction changes -
                                                                                             numerator negative and denominator positive
                                                                                             and fraction thus negative and sign changes.
  
  
  
  
  
                  2 ─ x            2 + x           1 ─ 6x
      19.   ──────  ━  ─────  ━  ───────
                 1 ─ 2x          1 + 2x          4x2 ─ 1
  
  
                        ─(─2 + x)            2 + x           ─(─1 + 6x)                 Verander die tekens
                  =  ────────  ━  ─────  ━  ─────────
                       ─(─1 + 2x)          1 + 2x          4x2 ─ 1                      Change the signs
  
  
                         x ─ 2          x + 2             6x ─ 1
                  =  ─────  ━  ─────  ┼  ──────
                        2x ─ 1        2x + 1          4x2 ─ 1
  
  
                         x ─ 2          x + 2                  6x ─ 1
                  =  ─────  ━  ─────  ┼  ───────────
                        2x ─ 1        2x + 1          (2x ─ 1)(2x + 1)
  
                         (x ─ 2)(2x + 1)  ─  (x + 2)(2x ─ 1) +  6x ─ 1
                  =  ─────────────────────────────
                                           (2x ─ 1)(2x + 1)
  
  
                         2x2 ─ 3x ─ 2  ─  (2x2 + 3x ─ 2) +  6x ─ 1
                  =  ─────────────────────────────
                                           (2x ─ 1)(2x + 1)
  
  
                         2x2 ─ 3x ─ 2  ─  2x2 ─ 3x + 2 +  6x ─ 1
                  =  ─────────────────────────────
                                           (2x ─ 1)(2x + 1)
  
  
                                  ─ 1
                  =  ───────────
                        (2x ─ 1)(2x + 1)
                                                                           Vr. / Qu. 19.
  
  
    OF / OR      
  
  
                  2 ─ x            2 + x           1 ─ 6x
      19.   ──────  ━  ─────  ━  ───────
                 1 ─ 2x          1 + 2x          4x2 ─ 1
  
  
                           2 ─ x                2 + x            1 ─ 6x                 Verander die tekens van een nie vier nie
                  =  ────────  ━  ─────  ━  ─────────
                           1 ─ 2x              1 + 2x        ─(─4x2 + 1)          Change the signs of one, not four.
  
  
                         2 ─ x          2 + x            1 ─ 6x
                  =  ─────  ━  ─────  ┼  ──────
                        1 ─ 2x        1 + 2x           1 ─ 4x2
  
  
                         2 ─ x          2 + x                  1 ─ 6x
                  =  ─────  ━  ─────  ┼  ───────────
                        1 ─ 2x        1 + 2x          (1 ─ 2x)(1 + 2x)
  
                         (2 ─ x)(1 + 2x)  ─  (2 + x)(1 ─ 2x) +  1 ─ 6x
                  =  ─────────────────────────────
                                           (1 ─ 2x)(1 + 2x)
  
  
                         2 + 3x ─ 2x2  ─  (2 ─ 3x ─ 2x2) +  1 ─ 6x
                  =  ─────────────────────────────
                                           (1 ─ 2x)(1 + 2x)
  
  
                         2 + 3x ─ 2x2  ─  2 + 3x + 2x2 +  1 ─ 6x
                  =  ─────────────────────────────
                                           (1 ─ 2x)(1 + 2x)
  
  
                                    1
                  =  ───────────
                        (1 ─ 2x)(1 + 2x)
                                                                           Vr. / Qu. 19.
  
  
                 3 ─ 2a          2a + 3              12
      20.   ──────  ━  ─────  ━  ───────
                 3 + 2a          2a ─ 3         4a2 ─ 9
  
  
                           3 ─ 2a             2a + 3                    12                    onnodig om tekens te verander
                  =  ────────  ━  ─────  ━  ─────────
                           2a + 3              2a ─ 3        (2a ─ 3)(2a + 3)          unnecessary to change any sign.
  
  
                         (3 ─ 2a)(2a ─ 3)  ─  (2a + 3)(2a + 3)  ─ 12
                  =  ─────────────────────────────
                                           (2a ─ 3)(2a + 3)
  
  
                         ─4a2 ─ 12a ─ 9  ─  4a2 ─ 12a ─ 9  ─ 12
                  =  ─────────────────────────────
                                           (2a ─ 3)(2a + 3)
  
  
                         ─8a2 ─ 24a ─ 30
                  =  ─────────────
                          (2a ─ 3)(2a + 3)                                                                           Vr. / Qu. 20.
  
  
  
  
                  3 ─ y            3 + y           1 ─ 16y
      21.   ──────  ━  ─────  ━  ───────
                 1 ─ 3y          1 + 3y          9y2 ─ 1
  
  
                           3 ─ y                3 + y            1 ─ 16y                 Verander die tekens
                  =  ────────  ━  ─────  ━  ─────────
                           1 ─ 3y              1 + 3y        ─(1 ─ 9y2)              Change the signs
  
  
                         3 ─ y          3 + y            1 ─ 16y
                  =  ─────  ━  ─────  ┼  ──────
                        1 ─ 3y        1 + 3y           1 ─ 9y2
  
  
                         (3 ─ y)(1 + 3y)  ─  (3 + y)(1 ─ 3y) +  1 ─ 16y
                  =  ─────────────────────────────
                                           (1 ─ 3y)(1 + 3y)
  
  
                         3 + 8y ─ 3y2  ─  3 + 8y + 3y2 +  1 ─ 16y
                  =  ─────────────────────────────
                                           (1 ─ 3y)(1 + 3y)
  
  
                                   1
                  =  ───────────
                        (1 ─ 3y)(1 + 3y)
                                                                           Vr. / Qu. 21.
  
  
                    3                   1                  3                 1
      22.   ──────  ━  ─────  ┼  ─────  ━  ─────
                 x + 1             x + 3            1 ─ x          3 ─ x
  
  
                          3                  1                    3                           1                 Verander die tekens
                  =  ─────  ━  ────  ┼  ────────  ─  ────────
                        x + 1           x + 3         ─(─1 + x)            ─(─3 + x)             Change the signs
  
  
                                                                                                                 Onthou die breuk is negatief en dus
                          3                  1            3                 1                                verander sy teken!
                  =  ─────  ━  ────  ─  ─────  ┼  ─────
                        x + 1           x + 3         x ─ 1           x ─ 3                         Remember the fraction is negative
                                                                                                                 and therefore its sign changes!
  
                         (3 ─ y)(1 + 3y)  ─  (3 + y)(1 ─ 3y) +  1 ─ 16y
                  =  ─────────────────────────────
                                           (1 ─ 3y)(1 + 3y)
  
  
                         3 + 8y ─ 3y2  ─  3 + 8y + 3y2 +  1 ─ 16y
                  =  ─────────────────────────────
                                           (1 ─ 3y)(1 + 3y)
  
  
                                   1
                  =  ───────────
                        (1 ─ 3y)(1 + 3y)
                                                                           Vr. / Qu. 22.
  
  
                 1 ─ 2a           1 + 2a           1 ─ 20a2
      23.   ──────  ━  ──────  ━  ───────
                 1 + 2a           1 ─ 2a           4a2 ─ 1
  
  
                         1 ─ 2a               1 + 2a            1 ─ 20a2                 Verander die tekens
                  =  ────────  ━  ─────  ━  ─────────
                         1 + 2a              1 ─ 2a            ─(1 ─ 4a2)               Change the signs
  
  
                                                                                                        Onthou die breuk is negatief en dus
                         1 ─ 2a               1 + 2a           1 ─ 20a2                  verander sy teken!
                  =  ────────  ━  ─────  ╋  ─────────
                         1 + 2a               1 ─ 2a           1 ─ 4a2                    Remember the fraction is negative and
                                                                                                        therefore its sign changes!
  
  
                         (1 ─ 2a)(1 ─ 2a)  ─  (1 + 2a)(1 + 2a) +  1 ─ 20a2
                  =  ─────────────────────────────
                                           (1 ─ 2a)(1 + 2a)
  
  
                         1 ─ 4a + 4a2  ─  1 ─ 4a ─ 4a2 +  1 ─ 20a2
                  =  ─────────────────────────────
                                           (1 ─ 2a)(1 + 2a)
  
  
                          1 ─ 8a ─ 20a2
                  =  ───────────
                        (1 ─ 2a)(1 + 2a)
  
  
                        (1 ─ 10a)(1 + 2a)
                  =  ────────────
                        (1 ─ 2a)(1 + 2a)
  
  
                        1 ─ 10a
                  =  ───────
                        1 ─ 2a                                                                           Vr. / Qu. 23.
  
  
                 a ─ 1              a + 1              4                   2
      24.   ──────  ━  ──────  ━  ─────  ┼  ─────
                 a ─ 2              a + 2           4 ─ a2          2 ─ a
  
  
                        a ─ 1           a + 1               4                           2                        Verander die tekens
                  =  ─────  ━  ────  ─  ────────  ┼  ────────
                        a ─ 2           a + 2         ─(─4 + a2)            ─(─2 + a)                Change the signs
  
  
                                                                                                                          Onthou die breuke is negatief
                        a ─ 1          a + 1                    4                          2                     en dus verander hulle tekens!
                  =  ─────  ━  ────  ┼  ───────────  ─  ─────
                        a ─ 2           a + 2         (a ─ 2)(a + 2)             a ─ 2                  Remember the fractions are
                                                                                                                          negative and therefore their
                                                                                                                          signs change!
  
  
                         (a ─ 1)(a + 2)  ─  (a + 1)(a ─ 2) +  4 ─ 2(a + 2)
                  =  ─────────────────────────────
                                           (a ─ 2)(a + 2)
  
  
                         a2 + a ─ 2  ─  a2 + a + 2 +  4 ─ 2a ─ 4
                  =  ─────────────────────────────
                                           (a ─ 2)(a + 2)
  
  
                                   0
                  =  ───────────       = 0
                        (a ─ 2)(a + 2)
                                                                           Vr. / Qu. 24.
  
  
                          5y                           15y ─ 15                       9y + 27
      25.   ──────────  ━  ────────────  ━  ────────────
                2y2 ─ 4y ─ 6           16y2 ─ 80y + 96           16y2 ─ 16y ─ 32
  
  
                                   5y                        15(y ─ 1)                   9(y + 3)
                  =  ──────────  ━  ───────────  ━  ──────────
                        2(y ─ 3)(y + 1)         16(y ─ 2)(y ─ 3)         16(y ─ 2)(y + 1)
  
  
                          5y X 8(y ─ 2)  ─  15(y ─ 1) X (y + 1)  ─  9(y + 3) X (y ─ 3)
                  =  ─────────────────────────────────────
                                                   16(y ─ 3)(y ─ 2)(y + 1)
  
  
                          40y2 ─ 80y  ─  15y2 + 15  ─  9y2 + 81
                  =  ──────────────────────────
                                   16(y ─ 3)(y ─ 2)(y + 1)
  
  
                           16y2 ─ 80y + 96
                  =  ─────────────────
                         16(y ─ 3)(y ─ 2)(y + 1)
  
  
                              16(y ─ 3)(y ─ 2)
                  =  ────────────────
                        16(y ─ 3)(y ─ 2)(y + 1)
  
  
                            1
                  =  ──────
                        (y + 1)                                                                                                     Vr. / Qu. 25.
  
  
  
                        c + 3d                           c + 2d                              c + d
      26.   ────────────  ┼  ────────────  ━  ─────────────
               4c2 + 12cd + 8d2           c2 + 4cd + 3d2             4c2 + 20cd + 24d2
  
  
                               c + 3d                           c + 2d                              c + d
                    =  ────────────  ┼  ────────────  ━  ─────────────
                          4(c2 + 3cd + 2d2)           c2 + 4cd + 3d2             4(c2 + 5cd + 6d2)
  
  
                               c + 3d                           c + 2d                              c + d
                    =  ────────────  ┼  ────────────  ━  ─────────────
                          4(c + d)(c + 2d)              (c + d)(c + 3d)               4(c + 2d)(c + 3d)
  
  
                          (c + 3d)(c + 3d)  +   (c + 2d) X 4(c + 2d)  ─ (c + d) X (c + d)
                    =  ─────────────────────────────────────
                                                4(c + d)(c + 2d)(c + 3d)
  
  
                          c2 + 6cd + 9d2  +   4c2 + 16cd + 16d2  ─ c2 ─ 2cd ─ d2
                    =  ─────────────────────────────────────
                                                4(c + d)(c + 2d)(c + 3d)
  
  
                              4c2 + 20cd + 24d2
                    =  ─────────────────
                          4(c + d)(c + 2d)(c + 3d)
  
  
                              4(c + 2d)(c + 3d)
                    =  ─────────────────
                          4(c + d)(c + 2d)(c + 3d)
  
  
                              1
                    =  ─────
                          c + d                                                                                                   Vr. / Qu. 26.
  
  
  
  
                        10b ─ 15                         7b                          36b + 12
      27.   ────────────  ┼  ──────────  ━  ─────────────
               66b2 + 11b ─ 11            6b2 + 7b ━ 3             44b2 + 88b + 33
  
  
                                5(2b ─ 3)                         7b                          12(3b + 1)
                    =  ────────────  ┼  ──────────  ━  ─────────────
                           11(6b2 + b ─ 1)            6b2 + 7b ━ 3             11(4b2 + 8b + 3)
  
  
                                  5(2b ─ 3)                         7b                              12(3b + 1)
                    =  ─────────────  ┼  ───────────  ━  ─────────────
                          11(2b + 1)(3b ─ 1)          (2b + 3)(3b ━ 1)           11(2b + 1)(2b + 3)
  
  
                           5(2b ─ 3) X (2b + 3)  +   7b X 11(2b + 1)  ─  12(3b + 1) X (3b ─ 1)
                    =  ────────────────────────────────────────────
                                                 11(2b + 1)(2b + 3)(3b ─ 1)
  
  
                           20b2 ─ 45 +  154b2 + 77b ─ 108b2 + 12
                    =  ───────────────────────────
                                       11(2b + 1)(2b + 3)(3b ─ 1)
  
  
                                   66b2 + 77b ─ 33
                    =  ──────────────────
                          11(2b + 1)(2b + 3)(3b ─ 1)
  
  
                                11(2b + 3)(3b ─ 1)
                    =  ──────────────────
                          11(2b + 1)(2b + 3)(3b ─ 1)
  
  
                                1
                    =  ──────
                          2b + 1                                                                                                   Vr. / Qu. 27.
  
  
  
  
                     3                       1                     a + 2b
      28.   ───────  ━  ───────  ┼  ─────────
                8a + 8b            8a ─ 8b            4(a2 + b2)
  
  
                               3                       1                     a + 2b
                   =  ───────  ━  ───────  ┼  ─────────
                         8(a + b)             8(a ─ b)            4(a2 + b2)
  
  
                         3(a ─ b)(a2 + b2)  ─   1(a + b)(a2 + b2)  +  (a + 2b) X 2(a + b)(a ─ b)
                   =  ───────────────────────────────────────────
                                                         8(a + b)(a ─ b)(a2 + b2)
  
  
                         3a3 + 3ab2 ─ 3a2b ─ 3b3  ─   a3 ─ ab2 ─ a2b ─ b3 + 2a3 ─ 2ab2 + 4a2b ─ 4b3
                   =  ──────────────────────────────────────────────────
                                                             8(a + b)(a ─ b)(a2 + b2)
  
  
                                   4a3 ─ 8b3
                   =  ────────────────
                         8(a + b)(a ─ b)(a2 + b2)
  
  
                          4(a3 ─ 2b3)
                   =  ──────────
                          8(a4 ─ b4)
  
  
                          a3 ─ 2b3
                   =  ──────────
                          2(a4 ─ b4)                                                                                             Vr. / Qu. 28.  
  
  
  
                     x3 + 8y3                    2x2 ─ 3xy ─ 2y2            2x2 + 5xy + 2y2
      29.   ───────────   X   ────────────   ÷   ───────────
                x2 ─ 3xy + 2y2              x2 ─ 2xy + 4y2                x2 ─ 2xy + y2
  
  
                                  (x + 2y)(x2 ─ 2xy + 4y2)             (x ─ 2y)(2x + y)                     (x ─ y)2
                            =  ────────────────   X   ────────────   X   ────────────
                                        (x ─ y)(x ─ 2y)                       x2 ─ 2xy + 4y2              (x + 2y)(2x + y)
  
  
                            =  x ─ y                                                                                              Vr. / Qu. 29.
  
  
  
                 x2 ─ 5x  + 6                x2 ─ 4x + 3                    x2 + 3x ─ 4
      30.   ──────────   ÷   ──────────   X   ───────────
                 x2 + 5x + 4                2x2 + 3x + 1                2x2 ─ 3x ─ 2
  
  
                                  (x ─ 2)(x ─ 3)             (x + 1)(2x + 1)             (x ─ 1)(x + 4)
                            =  ──────────   X   ──────────   X   ───────────
                                  (x + 1)(x + 4)              (x ─ 1)(x ─ 3)              (x ─ 2)(2x + 1)
  
  
                            =  1                                                                                                   Vr. / Qu. 30.
  
  
  
                   x2 + 5x  + 6                   x2 ─ 4x + 3                    x2 + 3x ─ 4
      31.   ───────────   ÷   ────────────   X   ───────────
                   x2 ─ 3x + 2                 x2 ─ 2xy + 4y2               x2 ─ 2xy + 4y2
  
  
                                  (x + 2)(x  + 3)             x2 ─ 2xy + 4y2            (x ─ 1)(x + 4)
                            =  ──────────   X   ──────────   X   ───────────
                                  (x ─ 1)(x ─ 2)             (x ─ 1)(x ─ 3)             x2 ─ 2xy + 4y2
  
  
                                  (x + 2)(x  + 3)(x + 4)
                            =  ───────────────
                                  (x ─ 1)(x ─ 2) (x ─ 3)                                                                Vr. / Qu. 31.
  
  
  
                1 + 8x3               4x ─ x3              (1 ─ 2x)2 + 2x
      32.   ───────   X   ───────   ÷   ───────────
                (2 ─ x)2              1 ─ 4x2              2 ─ 5x + 2x2
  
  
                                  (1 + 2x)(1 ─ 2x + 4x2)              x(2 ─ x)(2 + x)              1 ─ 2x + 4x2
                            =  ───────────────   X   ────────────   ÷   ───────────
                                          (2 ─ x)2                            (1 ─ 2x)(1 + 2x)            (x ─ 2)(2x ─ 1)
  
  
                                  (1 + 2x)(1 ─ 2x + 4x2)              x[─(x ─ 2)](x + 2)                (x ─ 2)(2x ─ 1)
                            =  ───────────────   X   ──────────────   X   ───────────
                                          [─(x ─ 2)]2                       [─(2x ─ 1)](2x + 1)                 1 ─ 2x + 4x2
  
  
                            =  x(x + 2)                                                                                 Vr. / Qu. 32.
  
  
  
                x2 + 2xy + y2 ─ a2            y2 ─ 2xy + x2 ─ c2            x + y + a
      33.   ─────────────   X   ─────────────   ÷   ───────
                y2 ─ c2 + 2cx ─ x2                (y ─ c)2 ─ x2                 y + x ─ c
  
  
                                  (x + y)2 ─ a2               (y ─ x)2 ─ c2            x + y + a
                            =  ──────────   X   ──────────   ÷   ───────
                                 y2 ─ (x ─ c)2                (y ─ c)2 ─ x2            x + y ─ c
  
  
                                   (x + y ─ a)(x + y + a)                 [─(x ─ y)]2 ─ c2                x + y ─ c
                            =  ───────────────   X   ──────────────   X   ───────
                                 [y ─ (x ─ c)][y + (x ─ c)]           (y ─ c ─ x)(y ─ c + x)            x + y + a
  
  
                                   (x + y ─ a)(x + y + a)             (x ─ y ─ c)(x ─ y + c)            x + y ─ c
                            =  ───────────────   X   ──────────────   X   ───────
                                 (y ─ x + c)(y + x ─ c)               (y ─ c ─ x)(y ─ c + x)            x + y + a
  
  
                                   (x + y ─ a)(x + y + a)             (x ─ y ─ c)(x ─ y + c)            x + y ─ c
                            =  ───────────────   X   ──────────────   X   ───────
                                  ─(x ─ y ─ c)(x + y ─ c)          ─(x ─ y + c)(x + y ─ c)           x + y + a
  
  
                                   (x + y ─ a)
                            =  ─────────
                                   (x + y ─ c)                                                                           Vr. / Qu. 33.
  
  
  
  
                4x2 + x ─ 14                   4x2            x ─ 2                   2x2 + 4x
      34.   ──────────   X   ──────  X  ──────   ÷   ──────────
                     6x ─ 14                  x2 ─ 4          4x ─ 7              3x2 ─ x ─ 14  
  
  
                                   (4x ─ 7)(x + 2)                     4x2                 x ─ 2              (x + 2)(3x ─ 7)
                              =  ──────────   X   ─────────  X  ──────   X   ──────────
                                        2(3x ─ 7)               (x ─ 2)(x + 2)          4x ─ 7                2x(x + 2)
  
  
                              =  x                                                                                         Vr. / Qu. 34.