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If the roots of the given equation are equal then discriminant is ZERO I.e.
( c - a )^2 - 4 ( b - c ) ( a - b ) = 0
===> c^2 + a^2 - 2ac + 4b^2 - 4ab + 4ac - 4bc = 0
===> c^2 + a^2 + 4b^2 + 2ac - 4ab - 4bc = 0
===> ( c + a - 2b ) ^2 = 0
===> c + a = 2b
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i hope this helps u.
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I hope you will understand by looking this answer