Question

How many values of c in the equation x^2-5x+c

Answer 1

Here we see the leading coefficient is 1

Coefficient of x= -5

The roots of the$x^2-5x+c$ can be obtained as follows:

$x=5\pm\frac{\sqrt{5^2-4(1)(c)}}{2(1)} =5\pm\frac{\sqrt{25-4c}}{2}$  

To get the roots as rational numbers, the value of  25-4c should be a perfect square

To get the roots as integers, the value $\sqrt{25-4c}$ should be an odd number

that is

$\sqrt{25-4c}=2n+1$ for some integer n

$25-4c=(2n+1)^2$  

$c=\frac{25-(2n+1)^2}{4}$  

Hence, we can conclude that c can be infinite as k can range over all integers