How many values of c in the equation x^2-5x+c result in the rational root…

Solution: Let r be an integer root of x^2 - 5x + c = 0.

Substitute x = r into the equation to get

r^2 - 5r + c = 0, so c = r(5 - r).

Because r can be any integer, the formula c = r(5 - r) produces infinitely many integer values of c.

• Examples: r = 0 gives c = 0; r = 1 gives c = 4; r = 2 gives c = 6; r = 6 gives c = -6.

• Note: values for r and 5 - r produce the same c (symmetry), but letting r vary over all integers still yields infinitely many distinct c values because c becomes arbitrarily negative as |r| grows.

Therefore the number of values of c that give an integer (hence rational) root is infinite.