Find all positive integer values of $c$ such that the equation $x^2-7x+c=0$ only has roots that are real and rational. Express them in decreasing order, separated by commas.
+128408 x^2 - 7x + c = 0
If tthe roots are real then this must be true
(-7)^2 - 4c > 0
49 - 4c > 0
49 > 4c
49/4 > c
The positive integer values that make this true are c = 1 - 12, inclusive
But....for rational solutions
49 - 4c must be a perfect square
So...the integer values that satisfy this are
c = 12, 10, 6
