Yes, that adapter is provided for this very reason, in case your PSU has no such connector. Use it and it'll work.
Your card requires this external power source because it needs more than the 75W the PCIe port can provide.
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CPU: AMD Phenom II X4 810 @ 3250MHz | RAM: Kingmax 2x2GB DDR2 800 @ 833MHz| MoBo: MSI K9A2 CF v1.0 (BIOS: 1.D)| GPU: Asus HD 6850 1024MB (DirectCu) @ 850/1100MHz | Display: Chili Green Vision L24FHD | PSU: PC Power & Cooling Silencer 750 Quad | OS: MS Windows 95 x64 Edition
CPU: Pentium 4 Northwood S478 @ 3200MHz | RAM: 1,5GB DDR 400| MoBo: Gigabyte GA-8S661FXMP-RZ | GPU: Abit Geforce Ti 4200 | Display: Dimarson 19" CRT | PSU: Noname 400W | OS: MS Windows XP Pro, Fedora 18