\(\text{For which positive integer values of $k$ does $kx^2+20x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order. }\)

Guest Aug 14, 2019

#1**0 **

\(\text{rational solutions means that the discriminant of the quadratic is a perfect square}\\ \text{for a quadratic $ax^2 + bx + c$ the discriminant}\\ D = b^2 - 4 a c \\~\\ D=(20)^2 - 4k^2 = 400- 4k^2 = 4(100-k^2)\\ \text{thus we need $100-k^2$ to be a perfect square}\\ \text{The only positive integer values of $k$ for which this is true are }\\ k=6, 8, 10\)

.Rom Aug 15, 2019