Consider the quadratic polynomial

MATHEMATICS · Subjective · AoPS · Mathematics · Quadratic Equations - Part 1 · Quadratic Equations - Part 1

Problem

q(x)=3x^{2}+kx+4

, where

k

is an integer. (i) Show that, for

q(x)

to split into two linear factors with integer coefficients, the number

k^{2}-48

must be a perfect square. (ii) Determine all integer values of

k

that satisfy the condition obtained in part (i). (iii) For each value of

k

found, factorise

q(x)

completely.