Expand: 14400/x^2+x^2/4=169.
Multiply through by 4x^2:
57600+x^4=676x^2.
x^4-676x^2+57600=0=(x^2-100)(x^2-576)
x^2=100 or 576. So x=±10 or ±24. Four solutions: x=-24, -10, 10, 24.
CHECK
±120/24=±5; ±24/2=±12 so 25+144=169=13^2 OK
±120/10=±12; ±10/2=±5 so 144+25=13^2 OK