WISKUNDE
GRAAD 11
NOG OEFENINGE
  
Oplos van kwadratiese vergelykings, vierkantsvergelykings : antwoorde.
  
MATHEMATICS
GRADE 11
MORE EXERCISES
  
Solving quadratic equations : answers.
  
  
             1.    x 2 + 4x + 3 = 0
  
                  (x + 1)(x + 3) = 0
  
                  (x + 1) = 0  OF / OR  (x + 3) = 0
  
                  x = ─ 1  OF / OR   x = ─ 3                     [ 1 ]
  
  
  
             3.   y =  1  OF / OR  y = 2                           [ 3 ]
  
  
  
             5.  a = ─ 2  OF / OR  a = 1                         [ 5 ]
  
  
  
             7.  p = ─ 1  OF / OR  p = 4                        [ 7 ]
  
  
  
             9.  x = ─ 3  OF / OR  x = 7                        [ 9 ]
  
  
  
  
  
                                                          5
           11.  q = ─ 5  OF / OR  p = ───
                                                          3                  [ 11 ]
  
  
  
  
  
                           1                                  6
           13.  z =  ──  OF / OR  z =  ─  ───
                           3                                  5
  
                                                                              [ 13 ]
  
  
  
                                 4                               6
           15.  b =  ─  ───  OF / OR  b =  ───
                                 7                               5
  
                                                                              [ 15 ]
  
  
  
                                                               
           17.  p =  ─ 5  OF / OR  p =  1
                                                               
  
                                                                              [ 17 ]
  
  
  
           19.  x2 + x ─ 3  =  0 
  
                     a = 1;  b = 1;  c = ─ 3
  
                                         _________
                             ─ b ± √ b2  ─  4ac
                   x =  ───── ─────────               
                                        2a
  
                                           ______________
                             ─ (1) ± √ (1)2  ─  4(1)(─3)
                      =  ───────────────────               
                                                 2(1)
  
                                         ____
  
                             ─ 1 ± √ 13
                      =  ──────────
                                     2
  
                                         ____
  
                             ─ 1 ─ √ 13
                   x =  ──────────
                                     2
  
                   x =  ─ 2,30     OF / OR
  
  
                                         ____
  
                             ─ 1 + √ 13
                   x =  ──────────
                                     2
  
  
                   x =  1,30                                               [ 19 ]
  
  
  
             21.  z = ─ 3,73  OF / OR  z = ─ 0,27
  
                                                                                [ 21 ]
  
  
  
             23.  b = ─ 7,53  OF / OR  b =  0,53
  
                                                                                [ 23 ]
  
  
  
             25.  b = ─ 0,79  OF / OR  b =  2,12
  
                                                                                [ 25 ]
  
  
  
             27.  x = ─ 0,40  OF / OR  b =  1
  
                                                                                [ 27 ]
  
  
  
             29.  z = ─ 2,52  OF / OR  z =  1,19
  
                                                                                [ 29 ]
  
  
  
             31.  q2 ─ 2q ─ 3  =  0
  
                         q2 ─ 2q    =  3
  
                                       1        ─ 2                    1        ─ 2
                   q2 ─ 2q + (── X ────)2  =  3+ (── X ────)2
                                       2         1                      2           1
  
                   q2 ─ 2q + 1  =  3 + 1
  
                   (q ─ 1)2  =  4
  
                   q ─ 1  =  ± √4  =  ± 2
  
                   q  = 1 ─ 2  OF / OR  q =  1 + 2
  
                   q  = ─ 1  OF / OR  q =  3
  
                                                                                [ 31 ]
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
             33.  y = ─ 2  OF / OR  y =  4
  
                                                                                [ 33 ]
  
  
  
             35.  a = ─ 0,85  OF / OR  a =  2,35
  
                                                                                [ 35 ]
  
  
  
             37.  p = ─ 4,54  OF / OR  p =  1,54
  
                                                                                [ 37 ]
  
  
  
             39.  x = ─ 1,00  OF / OR  x =  2,33
  
                                                                                [ 35 ]
  
  
  
          2.    x 2 + 7x + 10 = 0
  
              (x + 2)(x + 5) = 0
  
               (x + 2) = 0  OF / OR  (x + 5) = 0
  
               x = ─ 2  OF / OR   x = ─ 5                   [ 2 ]
  
  
  
           4.    y = 3  OF / OR  y = 7                        [ 4 ]
  
  
  
           6.    b = ─ 2  OF / OR  b = 3                    [ 6 ]
  
  
  
           8.    q = ─ 5  OF / OR  q = 3                    [ 8 ]
  
  
  
                                                              1
           10.  x = ─ 1  OF / OR  x =  ─  ───
                                                              2           [ 10 ]
  
  
  
                                3                                    2
           12.  z =  ─  ───  OF / OR  z =  ─  ───
                                2                                    3
  
                                                                          [ 12 ]
  
  
  
                                 4                               5
           14.  a =  ─  ───  OF / OR  a =  ───
                                 2                               3
  
                                                                          [ 14 ]
  
  
  
                                                               
           16.  y =  1  OF / OR  y =  2
                                                                
  
                                                                          [ 16 ]
  
  
  
                            5
           18.  y =  ───  OF / OR  y =  2
                            3
  
                                                                          [ 18 ]
  
  
  
           20.  y2 ─ y ─ 5  =  0 
  
                     a = 1;  b = ─ 1;  c = ─ 5
  
                                         _________
                             ─ b ± √ b2  ─  4ac
                   x =  ───── ─────────               
                                        2a
  
                                             ______________
                             ─ (─1) ± √ (─1)2  ─  4(1)(─5)
                      =  ───────────────────               
                                                 2(1)
  
                                       ____
  
                              1 ± √ 21
                      =  ──────────
                                     2
  
                                      ____
  
                             1 ─ √ 21
                   x =  ──────────
                                     2
  
                   x =  ─ 1,79     OF / OR
  
  
                                      ____
  
                             1 + √ 21
                   x =  ──────────
                                     2
  
  
                   x =  2,79                                           [ 20 ]
  
  
  
             22.  a =  0,44  OF / OR  a =  4,56
  
                                                                            [ 22 ]
  
  
  
             24.  a =  0,50  OF / OR  a =  1
  
                                                                            [ 24 ]
  
  
  
             26.  p = ─ 1,20  OF / OR  p =  1,45
  
                                                                            [ 26 ]
  
  
  
             28.  y =  1,29  OF / OR  p =  2,71
  
                                                                            [ 28 ]
  
  
  
             30.  a = ─ 1  OF / OR  a =  0,25
  
                                                                            [ 30 ]
  
  
  
             32.  x2 + 3x ─ 4  =  0
  
                         x2 + 3x    =  4
  
                                       1      3                  1        3
                   x2 ─ 3x + (── X ──)2 = 4 + (── X ──)2
                                       2      1                   2       1
  
                                       3                       3
                   x2 ─ 3x + (───)2  =  4 + (───)2
                                       2                       2
  
                            3                    9
                   (x ─ ──)2  =  4 + ───
                            2                    4
  
                            3             25
                   (x ─ ──)2  =  ───
                            2              4
  
                            3             5
                   x ─ ──  =  ± ───
                            2              2
  
                                      3          5
                             x  =  ──  ± ───
                                      2          2
  
                   x  = ─ 1  OF / OR  x =  4
  
                                                                                [ 32 ]
  
  
  
             34.  z = ─ 1  OF / OR  z =  0,67
  
                                                                            [ 34 ]
  
  
  
             36.  b = ─ 1,20  OF / OR  z =  1
  
                                                                            [ 36 ]
  
  
  
             38.  q = ─ 0,89  OF / OR  z =  3,39
  
                                                                            [ 38 ]
  
  
  
             40.  x = ─ 2  OF / OR  x =  4
  
                                                                            [ 40 ]