Question

find the roots of the quadratic equation (a-b)x2+(b-c)x+(c-a)=0 are equal , prove that 2a=b+c

Answer 1

$\underline\bold{\huge{SOLUTION \: :}}$

▶ Given, the roots of the given quadratic equation are equal.

▶ The general form of a quadratic equation is : ax² + bx + c = 0.

▶ Since, both the roots are equal, b²-4ac = 0.

▶ Now, we can proceed as under :

b² - 4 ac = 0

=> (b-c)² - 4 (a-b) (c-a) = 0

=> b²-2bc+c²-4ac+4a²+4bc-4ab = 0

=> 4a²+b²+c²-4ab+2bc-4ac = 0

=> (-2a)²+(b)²+(c)²+2(-2a) b+2(b)(c)+2(-2 a)c= 0

=> (-2a+b+c)² = 0

=> b+c-2a = 0

=> b+c = 2a

=> 2a = b+c [PROVED]

Answer 2

plz refer to this attachment