Hello everyone here on Math Help Forum.

I have an urgent question here: show that the equation $\displaystyle x^2 + kx = 4 - 2k $ has real roots for all real values of $\displaystyle k$.

What I know is that the term ''real roots'' imply that $\displaystyle b^2 - 4ac$ is equal or more than $\displaystyle 0$. But my primal problem is that I have difficulties in showing and proving. So here are my workings:

$\displaystyle x^2 + kx = 4 - 2k $

$\displaystyle x^2 + kx -4 +2k = 0$

Here $\displaystyle a = 1, b = k, c = -4 + 2k$

$\displaystyle b^2 -4ac = k^2 - 4(1)(-4+2k)$

$\displaystyle k^2 -8k +24$

$\displaystyle (x-4)(x-4)$

$\displaystyle (x-4)^2$

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!