If the roots of the equation x^2 - bx + c = 0 be two consecutive integers, then b^2 - 4c e…

Question

If the roots of the equation ${x^2} - bx + c = 0$ be two consecutive integers, then ${b^2} - 4c$ equals

Accepted Answer

Let $$\alpha ,\,\,\alpha  + 1\,\,$$ be roots

Then $$\alpha  + \alpha  + 1 = b = $$ sum of -

roots $$\alpha \left( {\alpha  + 1} \right) = c$$

$=$ product of roots

$$\therefore$$ ${b^2} - 4c$

$$ = {\left( {2\alpha  + 1} \right)^2} - 4\alpha \left( {\alpha  + 1} \right)$$

$= 1.$

Solutions · 1