Sir,For 2 Quadratic Equations what is the difference between the conditions when both have one and only one root in common and when both have atleast one root in common?Could you explain with suitable example and illustration?

If we take two quadtratic equations:

ax² + bx + c = 0

and

mx²+ nx + p = 0

Case I : Only One Common root

In this case Let that common root be ¥

Then

a¥² + b¥ + c = 0

and

m¥²+ n¥ + p = 0

Now add the two equations to get another quadratic equation:

(a+m)¥² + (b+n)¥ + c+p = 0

Replace ¥ by x

(a+m)x² + (b+n)x + c+p = 0.     Now this quadratic has only ine root

That means

D = 0

Or,

(b+n)² -4(a+m)(c+p) = 0

Case II : Atleast One common root:

Here D of [(a+m)x² + (b+n)x + c+p = 0. ]

D>=0

or

(b+n)² -4(a+m)(c+p) >=0