Mathematics

Problem

Let

a

be the rational number

4

and let

b = \sqrt{6}

. Consider the expression

P = (a + b)(a - b) + \dfrac{b^{2}}{a}

. a. (i) Simplify the expression

P

completely. b. (ii) Without approximating decimal values, determine whether the simplified value of

P

is rational or irrational. Give a clear justification. c. (iii) Replace the number

4

by a non-zero rational number

k

and write the corresponding expression

P_k = (k + b)(k - b) + \dfrac{b^{2}}{k}

. For what value(s) of

k

(if any) will

P_k

be an irrational number? Support your answer with reasoning.