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# Ôìóˆ ôè ïûâè ùë âíûˆûë âóè ôè

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∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 2 Î·È –1. R 1 = R 2 R R 2 – R . 1 R 1 ≠ 0. 1 R = 1 R 1 + 1 R 2 1 R 1 R 2 = 1 R 1 1 R 1 = R 2 – R R 2 R t = v – v 0 · 5 100 x + 3 100 (4000 – x) = 175 5x + 3(4000 – x) = 100 175 3 100 (4000 – x) 5 100 x ∞ª = 15 8 . 3(5 – x) 2 = 5x 2 15 – 3x = 5x 15 = 8x x = 15 8 . 3.1. ∂ÍÈÛÒÛÂÈ˜ 1Ô˘ ‚·ıÌÔ‡ 29

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8. i) x(x 2 – 1) – x 3 + x 2 = 0 x 3 – x – x 3 + x 2 = 0 x(x – 1) = 0 x = 0 x = 1. ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 0 Î·È 1. ii) (x + 1) 2 + x 2 – 1 = 0 x 2 + 2x + 1 + x 2 – 1 = 0 2x 2 + 2x = 0 2x(x + 1) x = –1 ‹ x = 0. ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› –1 Î·È 0. 9. i) x(x – 2) 2 = x 2 – 4x + 4 x(x – 2) 2 (x – 2) 2 = 0 (x – 2) 2 (x – 1) = 0 x – 2 = 0 x – 1 = 0 x = 2 x = 1. ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 2 Î·È 1. ii) (x 2 – 4)(x – 1) = (x 2 – 1)(x – 2) (x – 2)(x + 2)(x – 1) – (x – 1)(x + 1)(x – 2) = 0 (x – 1)(x – 2)[(x + 2) – (x + 1)] = 0 (x – 1)(x – 2) = 0 x = 1 x = 2 ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 1 Î·È 2. 10. i) x 3 – 2x 2 – x + 2 = 0 x 2 (x – 2) – (x – 2) = 0 (x – 2)(x 2 – 1) = 0 (x – 2)(x – 1)(x + 1) = 0 x – 2 = 0 x – 1 = 0 x + 1 = 0 x = 2 x = 1 x = –1. ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 2, 1 Î·È –1. ii) x 3 – 2x 2 – (2x – 1)(x – 2) = 0 x 2 (x – 2) – (2x – 1)(x – 2) = 0 (x – 2)(x 2 – 2x + 1) = 0 (x – 2)(x – 1) 2 = 0 x – 2 = 0 x – 1 = 0 x = 2 x = 1. ∂ÔÌ¤Óˆ˜ ÔÈ Ï‡ÛÂÈ˜ ÙË˜ ÂÍ›ÛˆÛË˜ Â›Ó·È ÔÈ ·ÚÈıÌÔ› 1 Î·È 2. ∫∂º∞§∞π√ 3: ∂•π™ø™∂π™ 30
11. i) ∏ ÂÍ›ÛˆÛË ·˘Ù‹ ÔÚ›˙ÂÙ·È ÁÈ· Î¿ıÂ x ≠ 1 Î·È x ≠ 0. ªÂ ·˘ÙÔ‡˜ ÙÔ˘˜ ÂÚÈÔÚÈÛÌÔ‡˜ ¤¯Ô˘ÌÂ: x = –1 (·ÊÔ‡ x ≠ 1). ∂ÔÌ¤Óˆ˜ Ë ÂÍ›ÛˆÛË ¤¯ÂÈ ÌÔÓ·‰ÈÎ‹ Ï‡ÛË ÙËÓ x = –1. ii) ∏ ÂÍ›ÛˆÛË ·˘Ù‹ ÔÚ›˙ÂÙ·È ÁÈ· Î¿ıÂ x ≠ 1 Î·È x ≠ –1. ªÂ ·˘ÙÔ‡˜ ÙÔ˘˜ ÂÚÈÔÚÈÛÌÔ‡˜ ¤¯Ô˘ÌÂ: x – 1 + 2 = 0 x + 1 = 0 x = –1, Ô˘ ·ÔÚÚ›ÙÂÙ·È ÏfiÁˆ ÙˆÓ ÂÚÈÔÚÈÛÌÒÓ. ∂ÔÌ¤Óˆ˜ Î·È Ë ·Ú¯ÈÎ‹ ÂÍ›ÛˆÛË Â›Ó·È ·‰‡Ó·ÙË. 12. i) ∏ ÂÍ›ÛˆÛË ·˘Ù‹ ÔÚ›˙ÂÙ·È ÁÈ· Î¿ıÂ x ≠ 1 Î·È x ≠ –1. ªÂ ·˘ÙÔ‡˜ ÙÔ˘ ÂÚÈÔÚÈÛÌÔ‡˜ ¤¯Ô˘ÌÂ: x + 1 + x – 1 = 2 2x = 2 x = 1, Ô˘ ·ÔÚÚ›ÙÂÙ·È, ·ÊÔ‡ x ≠ 1.

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