Question

If the roots of $x^2-bx+c=0$are two consecutive integers, then $b^2-4c$is

• A. 0
• B. 1
• C. 2
• D. none of these

Answer 1

The correct option is B

1

Given equation: $x^2-bx+c=0$

Let $\alpha and\alpha +1$ be the two consecutive roots of the equation.

Sum of the roots = $\alpha +\alpha +1=2\alpha +1$

Product of the roots = $\alpha (\alpha +1)={\alpha }^2+\alpha$

So, sum of the roots = $2\alpha +1$ $=\frac{-Coefficientofx}{Coefficientof{x}^2}=\frac{b}{1}=b$

Product of the roots $={\alpha }^2+\alpha =\frac{Constantterm}{Coeffecientof{x}^2}=\frac{c}{1}=c$

Now, $b^2-4c=(2\alpha +1{)}^2-4({\alpha }^2+\alpha )=4{\alpha }^2+4\alpha +1-4{\alpha }^2-4\alpha =1$