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

Let the two consecutive integer roots of the equation x
2
−bx+c=0 be α and α+1.

From the properties of a quadratic equation:

Sum of roots:

α+(α+1)=b⟹2α+1=b
Product of roots:

α(α+1)=c⟹α
2
+α=c
We need to find the value of the discriminant expression b
2
−4c:
Substitute the values of b and c in terms of α:

b
2
−4c=(2α+1)
2
−4(α
2
+α)
Expanding the terms:

b
2
−4c=(4α
2
+4α+1)−(4α
2
+4α)
b
2
−4c=4α
2
+4α+1−4α
2
−4α
b
2
−4c=1
Alternatively, since the roots are consecutive integers, their difference is always ∣α−β∣=1.
The formula for the difference of roots is:

∣α−β∣=
∣a∣
b
2
−4ac

​

​

Given a=1:

1=
1
b
2
−4c

​

​
⟹1=b
2
−4c