simple though interesting...!!

They say for a quadratic eqn. 'ax² + bx + c = 0' to have integer roots 'a' must equal 1 and the roots must be rational.

Prove that when this happens the roots will have be integers...!!

2 Answers

Typo: 2 is missing in the last line after have... :P

well no one here for this.........!!

262

Aditya Bhutra ·2012-03-13 00:00:50

eqn reduces to x² +bx+c =0

roots are x =-b Â± √b²-4c2

case 1 : b =2n

then b²-4c = 4k² (k² since roots are rational)

x=-2n Â± 2k2 = kÂ±n (integer)

case 2 : b=2n+1

then b²-4c = odd = (2k+1)²

x=-(2n+1) Â± (2k+1)2 = k-n , -n-k-1 (integer)

hence proved.

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