Question

If the roots of the equation (a - b)X + (b-c)x+ (c-a)=0 are equal,show thatc,a and b are in AP.​​​.

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Answer 1

Answer:

ac square = bc square + ab square

Step-by-step explanation:

If the roots of the equation (a - b)X + (b-c)x+ (c-a)=0 are equal,show thatc,a and b are in AP..

Answer 2

The question must be, if roots of equation (a-b)x2+(b-c)x+(c-a)=0 are equal , prove that b+c = 2a.

Using Discriminant,

D = B2-4AC as compared with the general quadratic equation Ax2+Bx+C=0

so, A = a-b

B = b-c

C = c-a

For roots to be equal, D=0

(b-c)2 - 4(a-b)(c-a) =0

b2+c2-2bc -4(ac-a2-bc+ab) =0

b2+c2-2bc -4ac+4a2+4bc-4ab=0

4a2+b2+c2+2bc-4ab-4ac=0

(2a-b-c)2=0

i.e. 2a-b-c =0

2a= b+c