Showing that the equation has real roots for all real values of 'k'

Hello everyone here on Math Help Forum.

I have an urgent question here: show that the equation has real roots for all real values of .

What I know is that the term ''real roots'' imply that is equal or more than . But my primal problem is that I have difficulties in showing and proving. So here are my workings:

Here

I have tried my best. Please pinpoint my mistakes and kindly continue the rest of the question. Thank you so much for your help and have a nice day!