Roots of equation x^2 - rx + m = 0 differ by 1. Prove that r^2 = 4m + 1

The equation is x^2 - rx + m = 0 and the difference between its roots is 1.

We know that for the equation ax^2 + bx + c = 0, 

Difference of roots = √D /a 

where D is the Discriminant of the equation, which equals (b^2 - 4ac). So, here,

Difference of roots = √[(-r)^2 - 4(1)(m)] / 1

=> 1 = √(r^2 - 4m)

Squaring both the sides,

=> 1 = r^2 - 4m

=> r^2 = 4m + 1

Hence Proved