Let be an integer and consider the quadratic expression .

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let

k

be an integer and consider the quadratic expression

2x^{2}+kx+3

. (i) Determine all integer values of

k

for which the expression can be factorised completely over the integers. (ii) For every value of

k

found in part (i), write the explicit factorised form. (iii) From the values of

k

obtained, identify those for which both roots of the equation

2x^{2}+kx+3=0

are positive integers and justify your answer.