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# 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?

12 years ago

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

2 + b¥ + c = 0

and

2+ n¥ + p = 0

(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