Question

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

• A. $-1$
• B. $0$
• C. $1$
• D. $2$
• E. $3$

Answer 1

The correct option is C

$1$

Explanation of correct option.

Quadratic equation:

Let $\alpha ,\alpha +1$be the two consecutive integers.

$$\begin{gathered} \therefore (x+\alpha )(x+\alpha +1)=x^2-bx+c \\ \Rightarrow x^2+x\alpha +x+\alpha x+{\alpha }^2+\alpha =x^2-bx+c \\ \Rightarrow x^2+2x\alpha +x+{\alpha }^2+\alpha =x^2-bx+c \\ \Rightarrow x^2+x(2\alpha +1)+{\alpha }^2+\alpha =x^2-bx+c \end{gathered}$$

On comparing,

$$\begin{gathered} -b=2\alpha +1 \\ c={\alpha }^2+\alpha \end{gathered}$$

Now,

$$\begin{gathered} b^2-4c=(2a+1{)}^2-4\left({\alpha }^2+\alpha \right) \\ =1 \end{gathered}$$

Hence, the correct answer is an option (C)