How do I show this quantity is greater than or equal to $0$?

Question

I am trying to show that the quantity under a radical is equal to $0$ or positive in order to show my roots must be real.

I have simplified my radical to this: $$\sqrt{a^2+4b^2+d^2-2ad}$$I allow $a,b,d$ to vary across the real numbers. How can I show $a^2+4b^2+d^2-2ad\geq0$?

Answer 1

It is $$a^2-2ad+d^2+4b^2=(a-d)^2+4b^2\geq 4b^2\geq 0$$

Answer 2

Hint: $(a-d)^2=a^2-2ad+d^2$ Try using this

Answer 3

Recall :

1)$(a+d)^2 \ge 0;$ hence $a^2+d^2 \ge - 2ad.$

2)$ (a-d)^2 \ge 0$; hence $a^2+d^2 \ge 2ad.$

Together : $a^2+d^2 \ge 2|ad|$.

Does this help?