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

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

(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.

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

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

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

(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.

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 .

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 .

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 .