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Q:

Given that  is a quadratic equation, find the value(s) of the constant  for which
the roots are equal.

A:

For equal roots

 or

Year: June 2021 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

    Q:

    Given that  is a quadratic equation, find the value(s) of the constant  for which
    the roots are equal.

    A:

    For equal roots

     or

    Year: June 2021 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

      Q:

      (a) The roots of the quadratic equation  are  and . Find the quadratic equation with integral coefficients whose roots are  and
      (b) Find the value of the constant        for which the equation  has imaginary roots.

      A:

      a)            Sr;  and Pr; ()=

      New roots  and

      Snr; ) +(  =( =  -2()= -

      And Pnr; )( =(  )-2()+9

      b)           

      For imaginary roots

      Year: June 2019 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

        Q:

        Given that one root of the quadratic equation  is three times the other, find the value of the constant . Hence, solve the equation

        A:

        Let roots be  and
         and  but
         or

        Year: June 2018 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

          Q:

          Find the set of values of  for which the roots of the quadratic equation  are real and different

          A:

          We have . for real and distinct roots

          Year: June 2017 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

            Q:

            The roots of the quadratic equation  are  and . Prove that:
            a)
            b)

            Hence, form a quadratic equation whose roots are  and
            c) Find the range of values of             , for which

            A:

            Given equation , with roots  and
            a)
            bi
            New equation:
            c) For , we have
             the zeros are:  and

            Possible range

            +++

            +++

            +++

            -

            +++

            +++

            ++

            Thus, range

            Year: June 2014 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

              Q:

              (i) Given that the roots of the equation  are  and , form the quadratic equation whose roots are  and .
              (ii) Find the values) of the constant               , for which the roots of the equation  are equal.

              A:

              Equation  have roots  and
               and
               and

              New equation:
              (ii) If the roots of

              Year: June 2012 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                Q:

                Given that p and q are the roots of the equation , where a is a real constant, show that an equation whose roots are  and  is .

                A:


                 and
                 and
                New equation:


                Year: June 2011 | Subject: Mathematics | Topic: POLYNOMIALS

                  Q:

                  Given that  and  are the roots of the equation , where n is a constant, find, in terms of n , the quadratic equation whose roots are  and .

                  A:



                  New roots:  and
                  and
                  New equation:

                  Year: June 2008 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                    Q:

                    Given that the roots of the quadratic equation , are real and equal, find the values of the real constants a. Using the smaller value of a form a quadratic equation whose roots are  and  where  is the root of the equation .

                    A:

                    For real roots

                     or 18 Smaller value of


                    New roots  and
                     and  New Equation

                    Year: June 2007 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                      Q:

                      i) If the roots of the quadratic equation  are  and . Find a quadratic equation whose roots are  and .
                      (ii) Given that                       , find the two values of p for which the equation  has real
                      roots.

                      A:

                      i. Equation:  has roots  and  and

                      Now roots:  and   new equation:
                      ii. If                        or  has real and equal roots, then  or

                      Year: June 2002 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                        Q:

                        Given that a is a real constant, prove that the roots  and  of the quadratic equation
                         are real. Find the value of a in the case where .

                        A:

                        We have
                        For real roots,
                        Completing the squares, gives
                        Since  and
                        Hence , has real roots  and
                        In the case where  for the equation , we have ;
                         

                        and .
                        from (1),  substituting in (2)

                        Year: June 1999 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                          Q:

                          Given that  and  are the roots of the equation , where b is a constant, find the quadratic equation whose roots are  and .

                          A:



                          Now equation

                          Year: June 1998 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                            Q:

                            The roots of the equation  are  and . Form a quadratic equation with roots  and .

                            A:

                            Sr :

                            New roots  and

                            new equation

                            Year: June 1996 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                              Q:

                              The equation  has roots  and . Find without solving this equation, the values of  for which
                              a)
                              b)                               .

                              A:

                              a)  but  or  (1)
                               but  or  or

                              b) Sr :  but   or
                              Pnr :  but  or


                              Year: June 1995 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                                Q:

                                (i) Find the range of values of  for which the equation  has
                                a) Real roots
                                b) Imaginary roots.
                                (ii) The roots of the equation                                  are  and . Without solving this equation form the quadratic equation whose roots are  and .

                                A:

                                (i) (a) for real roots
                                 or
                                (b) For imaginary roots

                                (ii)

                                New roots

                                New equation

                                Year: June 1994 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                                  Q:

                                  Show that if  and  are the roots of the quadratic equation , where  are real constants then . The roots of the quadratic equation , where b is a real constant are .  and . Find in terms of , the quadratic equation whose roots are  and  where , . Hence, find the range of values of b for which  and  are real and distinct.

                                  A:

                                  Given that  and  are roots then  and  are factors of

                                  Comparing coefficients gives  and
                                  Given the equation  with roots  and  and
                                  New roots are  and

                                  New equation
                                  Range of values of  for real roots implies

                                  Critical values are

                                  Possible range

                                  +++

                                  +++

                                  +++

                                  +++

                                  +++

                                  --

                                  --

                                  ---

                                  +++

                                  +++

                                  -

                                  +++

                                  Range of values of 

                                  Year: June 1992 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                                    Q:

                                    Given that  and  are roots of an equation  where , show that

                                    A:

                                    Year: June 1991 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                                      Q:

                                        Given that  are the roots of the equation . Find the quadratic equation whose
                                      roots are
                                      a)                                        and
                                      b)  and .

                                      A:

                                      Year: june 1989 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

                                        Q:

                                        Given that the roots of the equation  arc  and , form the equation whose roots are  and . Find the values of r for which  and  are equal.

                                        A:

                                         and  new roots  and
                                         and

                                        new equation:


                                        For equal roots

                                        Year: june 1983 | Subject: Mathematics | Topic: QUADRATIC EQUATIONS

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