Roots of a Quadratic Equation.

Algebra Level 3

If the roots of $x^2 +bx +c = 0$ are two consecutive integers. What is the value of $b^2-4c -1=?$

2 0 4 1 3

1 solution

Hana Wehbi
Jul 2, 2017

Let $n$ and $n+1$ to be the two consecutive roots.

Then Sum of the roots is $\frac{-b}{a} = \frac{-(2n+ 1)}{1}\implies b^2= (2n+1)^2= 4n^2+4n+1$

While the Product of the roots is $\frac{c }{a}=n^2 + n \implies 4c= 4n^2+ 4n$

Now: $b^2 - 4c -1= 4n^2 +4n +1 - 4n^2 -4n -1= 0$ .

In my solution, l used Vieta's Formula for Quadratics.