Question

What is the condition for equation,x^2-bx+c=0(b&c are integers)to have two consecutive integers as it's roots?
A: b^2 -c^2=1
B: b^2 -4c=1
C: c^2 -4b^2=1
D: b=c

Answer 1

let the roots be α and α+1

then.,

α+α+1=b
b=2α+1.......(1)
α=(b-1)/2

and
α(α+1)=c
${ \alpha }^{2} + \alpha = c....(2)$
from (2) and (1)
${ (\frac{b - 1}{2}) }^{2} + ( \frac{b - 1}{2} ) = c$
$\frac{b - 1}{2} ( \frac{b - 1}{2} + 1) = c \\ ( \frac{b - 1}{2} )( \frac{b + 1}{2} ) = c \\ {b}^{2} - 1 = 4c$
${b}^{2} - 4c = 1$
option (c)


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