Question

Find a relation between a, b and c if the roots of the equation (b – c) x² + (c – a) x + (a – b) =

0 are equal.​

Answer 1

$\setlength{\unitlength}{1cm}\begin{picture}(6, 6) \put(2, 3){\vector(1, 0){3}}\end{picture}$  QUESTION The question must be, if roots of equation (a-b)x2+(b-c)x+(c-a)=0 are equal , prove that b+c = 2a.

$\setlength{\unitlength}{1cm}\begin{picture}(6, 6) \put(2, 3){\vector(1, 0){3}}\end{picture}$ ANSWER

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

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