Mathematics

Problem

Study the quadratic equation

(a - 1)x^2 + 2(a + 1)x + (a - 3) = 0

, where

a

is a real parameter and

a \neq 1

. a. (i) Obtain an expression for the discriminant in terms of

a

. b. (ii) Hence determine the set of values of

a

for which the roots are (a) equal and real, (b) real and distinct, (c) non-real. c. (iii) Taking

a = 2

as a particular case, calculate the roots explicitly and then draw a qualitative sketch of the graph of

y = (a - 1)x^2 + 2(a + 1)x + (a - 3)

for this value of

a

, marking intercepts and vertex.