Question

the solution for this question is​

Answer 1

1. Given quadratic equation is (a – b)x2 + (b – c)x + (c – a) = 0

Since the root are equal, discriminant of the quadratic equation = 0

Hence (b – c)2 = 4(a – b)(c – a)

⇒ b2 + c2 – 2bc = 4(ac – a2 – bc + ab)

⇒ b2 + c2 – 2bc = 4ac – 4a2 – 4bc + 4ab

⇒ b2 + c2 – 2bc – 4ac + 4a2 + 4bc – 4ab = 0

⇒ b2 + c2 + 4a2 + 2bc – 4ac – 4ab = 0

⇒ b2 + c2 + (2a)2 + 2(b)(c) – 2(2a)c – 2(2a)b = 0

⇒ b2 + c2 + (–2a)2 + 2(b)(c) + 2(–2a)c + 2(–2a)b = 0

⇒ (b + c – 2a)2 = 0

⇒ b + c – 2a = 0

∴ b + c = 2a

2.  ( 1/ [a+b+x] ) = (1/a) +( 1/b )+ (1/x)

( 1 / [a+b+x] ) - (1 / x ) = (1/a) + (1/b)

=> { x - [a+b+x] } / ([a+b+x] * x ) = {a+b} / ab

=> - { a+b} / ( [a+b+x] * x ) = { a+b} / ab

=> -1 / ( [a+b+x] * x ) = 1 / ab

Cross Multiply : -ab = [a+b+x] * x

On Simplification : x2 + (a+b) x + ab = 0

Applying the Splitting the middle term method :

=> x2 + (a+b) x + ab = 0

=> [x2 + ax]+ [b x + ab] = 0

So : x ( x + a) + b ( x + a) = 0

=> (x + a ) * ( x + b ) = 0

Therefore : (x+a) = 0 or (x+b) = 0

Now : x = -a or x = -b

The value of x is : -a or -b