Mathematics

Problem

For the quadratic equation

kx^{2}+(k-3)x+2=0

, determine all real values of

k

that satisfy the following conditions simultaneously: (i) One of the zeros is

x=2

. (ii) The other zero is equal to the reciprocal of

2

. (a) Use condition (i) to obtain a linear equation in

k

and simplify it. (b) Apply condition (ii) to the same quadratic equation and obtain a second linear equation in

k

. (c) Solve the two linear equations obtained in parts (a) and (b) simultaneously to find the permissible value(s) of

k

. (d) For each value of

k

obtained, write the corresponding quadratic equation and verify explicitly that its zeros satisfy conditions (i) and (ii).