Equation (a - b)x^2 - (b + c - a)x - c = 0, find its nature of roots.

The equation given is (a - b)x^2 - (b + c - a)x - c = 0. To determine the nature of roots, we'll first find the discriminant of the equation. We know that for ax^2 + bx + c = 0, discriminant D = b^2 - 4ac

So, here,

D

= [-(b + c - a)]^2 - 4(a - b)(-c)

= b^2 + c^2 + a^2 + 2bc - 2ac - 2ab + 4ac - 4bc

= b^2 + c^2 + a^2 - 2bc + 2ac - 2ab

If you look at this expression carefully, it is in form of a perfect square.

= (-b)^2 + (c)^2 + (a)^2 + 2(-b)(c) + 2(c)(a) + 2(a)(-b)

= (-b + c + a)^2

So, D = (-b + c + a)^2 for the given equation. Lets make our conclusions now.

Since D is a perfect square, the equation will have rational roots. 

Again, since (-b + c + a)^2 is a perfect square, it is always greater than or equal to 0. The roots will be real. (of course, rational numbers are always real)

So, we can say that the equation (a - b)x^2 - (b + c - a)x - c = 0 will have real and rational roots for all real values of k.