2011 &Icirc;›&Iuml;&Iuml;ƒ&Icirc;&micro;&Icirc;&sup1;&Iuml;‚ &Icirc;&pound;&Iuml;‡. &Icirc;&sup2;&I

# Óè 2 2 2 4 2 2 4 4 2 2 2 4 2 2 4 4 2 2 2 2 2 2 ñ

• Notes
• 120

This preview shows 44 out of 46 pages.

∂›Ó·È ¢ = 2 + ‚ 2 ) 2 – 4· 2 2 = · 4 + ‚ 4 + 2· 2 2 – 4· 2 2 = · 4 + ‚ 4 – 2· 2 2 = (· 2 – ‚ 2 ) 2 ñ ∞Ó · ≠ ±‚ Ë ÂÍ›ÛˆÛË ¤¯ÂÈ ‰‡Ô Ú›˙Â˜ ÙÈ˜ ñ ∞Ó · = ‚ ‹ · = –‚ ÙfiÙÂ Ë ÂÍ›ÛˆÛË ¤¯ÂÈ ‰ÈÏ‹ Ú›˙·, ÙËÓ 6. i) Œ¯Ô˘ÌÂ ¢ = 4Ï 2 – 4 (–8) = 4Ï 2 + 32 > 0 ÁÈ· Î¿ıÂ Ï . ∞˘Ùfi ÛËÌ·›ÓÂÈ fiÙÈ Ë ÂÍ›- ÛˆÛË ¤¯ÂÈ Ú›˙Â˜ Ú·ÁÌ·ÙÈÎ¤˜ ÁÈ· Î¿ıÂ Ï . x = · 2 + ‚ 2 2‚ = 2 2‚ = · 2 = ± 2 = ‚. x 2 = · 2 + ‚ 2 – · 2 + ‚ 2 2‚ = 2‚ 2 2‚ = ‚. x 1 = · 2 + ‚ 2 + · 2 – ‚ 2 2‚ = 2 2‚ = · 2 Î·È x · + · x = · + · ·‚x x · + ·‚x · x = ·‚x · + ·‚x · x · + · x = · + · ·‚x x · + ·‚x · x = ·‚x · + ·‚x · x = ‚ x = · 2 . x = ‚ · ‚x = 1 · x = ‚ ‚x = · 2 ∫∂º∞§∞π√ 3: ∂•π™ø™∂π™ 44

Subscribe to view the full document.

ii) ŒÛÙˆ x 1 , x 2 ÔÈ Ú›˙Â˜ ÙË˜ ÂÍ›ÛˆÛË˜ ÌÂ x 2 = x 1 2 . ∞fi ÙÔ˘˜ Ù‡Ô˘˜ Vieta ¤¯Ô˘ÌÂ ñ x 1 + x 2 = –2Ï x 1 + x 1 2 = –2Ï Î·È ñ x 1 Ø x 2 = –8 x 1 3 = –8 x 1 = –2, ÔfiÙÂ x 2 = (–2) 2 = 4. ΔfiÙÂ ¤¯Ô˘ÌÂ –2 + 4 = –2Ï 2Ï = –2 Ï = –1. 7. ŒÛÙˆ x – 1, x, x + 1 ÙÚÂÈ˜ ‰È·‰Ô¯ÈÎÔ› ·Î¤Ú·ÈÔÈ. √È ·ÚÈıÌÔ› ·˘ÙÔ› ·ÔÙÂ- ÏÔ‡Ó ÏÂ˘Ú¤˜ ÔÚıÔÁˆÓ›Ô˘ ÙÚÈÁÒÓÔ˘ ·Ó Î·È ÌfiÓÔ ·Ó ÈÛ¯‡ÂÈ (x + 1) 2 = x 2 + (x – 1) 2 x 2 + 2x + 1 = x 2 + x 2 – 2x + 1 x 2 – 4x = 0 x(x – 4) = 0 x = 4, ·ÊÔ‡ x ≠ 0 ˆ˜ ÏÂ˘Ú¿ ÙÚÈÁÒÓÔ˘. ∏ Ï‡ÛË x = 4 ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÌÔÓ·‰ÈÎ‹. ∂ÔÌ¤Óˆ˜ ˘¿Ú¯ÂÈ Ì›· ÌfiÓÔ ÙÚÈ¿‰· ‰È·‰Ô¯ÈÎÒÓ ·ÎÂÚ·›ˆÓ Ô˘ Â›Ó·È Ì‹ÎË ÏÂ˘ÚÒÓ ÔÚıÔÁˆÓ›Ô˘ ÙÚÈÁÒ- ÓÔ˘. √È ·Î¤Ú·ÈÔÈ ·˘ÙÔ› Â›Ó·È ÔÈ 3, 4 Î·È 5. 8. ΔÔ ÂÌ‚·‰fiÓ ∂ 1 ÙÔ˘ ÛÙ·˘ÚÔ‡ ÚÔÎ‡ÙÂÈ ·fi ÙÔ ¿ıÚÔÈÛÌ· ÙˆÓ ÂÌ‚·‰ÒÓ ÙˆÓ ‰‡Ô ÏÂ˘ÎÒÓ ÏˆÚ›‰ˆÓ ÙË˜ ÛËÌ·›·˜ ·fi ÙÔ ÔÔ›Ô fiÌˆ˜ Ú¤ÂÈ Ó· ·Ê·ÈÚ¤ÛÔ˘ÌÂ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÎÔÈÓÔ‡ ÙÂÙÚ·ÁÒÓÔ˘ (√ªπ∑) ÏÂ˘Ú¿˜ d. ∂›Ó·È ‰ËÏ·‰‹ 1 = 3 Ø d + 4 Ø d – d 2 = 7d – d 2 ŒÛÙˆ ∂ 2 ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ˘fiÏÔÈÔ˘ Ì¤- ÚÔ˘˜ ÙË˜ ÛËÌ·›·˜. £· ÈÛ¯‡ÂÈ ∂ 1 = ∂ 2 ·Ó Î·È ÌfiÓÔ ·Ó ÙÔ ∂ 1 Â›Ó·È ›ÛÔ ÌÂ ÙÔ ÌÈÛfi ÙÔ˘ ÂÌ‚·‰Ô‡ ÔÏfiÎÏËÚË˜ ÙË˜ ÛËÌ·›- ·˜. ∂ÔÌ¤Óˆ˜ ¤¯Ô˘ÌÂ 1 = ∂ 2 7d – d 2 = 3 Ø 4 2 d 2 – 7d + 6 = 0 d = 1 d = 6.
You've reached the end of this preview.
• Winter '09
• Nikos

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern