Mathematics

Problem

Let

k

be an integer and consider the quadratic expression

2x^{2}+kx+3

. a. (i) Determine all integer values of

k

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

k

found in part (i), write the explicit factorised form. c. (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.

Intuitive explanation

The key is recognizing that a quadratic with integer coefficients factors nicely over the integers if and only if its discriminant is a perfect square. Here, you'll compute $\Delta = k^2 - 24$ and determine which integer values of $k$ make this a perfect square—that's what forces the quadratic to split into linear factors with rational (actually integer) roots.

Intuitive explanation

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