<s>[INST] Prove that roots are real I am stuck with this equation, I need to prove the roots are real when $a, b, c \in R$
The equation is $$(a+b-c)x^2+ 2(a+b)x + (a+b+c) = 0$$
If someone could tell me the right way to go about this, so I can attempt it.
Thank you
EDIT: I have made an error in the question.  I have now corrected it.
 [/INST] $$ax^2 +bx^2 + cx^2 + 2ax+2bx + a+b+c = 0$$
or
$$x^2(a+b-c) + 2x(a+b) +a+b+c = 0$$
Let us now find the expression for the discriminant,
$$\Delta = 4(a+b)^2-4(a+b-c)(a+b+c)$$
If we prove that, $\Delta\geq 0$, we done.
$$4(a+b)^2-4(a+b+c)(a+b-c)=$$
$$=4a^2+8ab+4b^2-4(a^2+2ab+b^2-c^2)=$$
$$=4a^2+8ab+4b^2-4a^2-8ab-4b^2+4c^2=$$
$$4c^2\geq 0$$
Hence, indeed
$$\Delta\geq 0$$
and therefore the quadratic equation has real roots.
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