Вправи 401 - 467 » 411

411. Обчисліть а2 + а–2, якщо: 1) якщо a + a–1 = 5, то a2 + a–2 = a2 + (a–1)2 + 2a • a–1 – 2a • a–1 = (a + a–1)2 – 2a • a–1 = 52 – 2 • 1 = 25 – 2 = 23. 2) якщо a – a–1 = 1, то a2 + a–2 = a2 + (a–1)2 + 2aa–1 + 2aa–1 = (a – a–1)2 + 2a • a–1 = 12 + 2 • 1 = 1 + 2 = 3. 3) якщо a + 1/2 = 2, то a2 + a–2 = a2 + (a–1)2 + 2a • a–1 – 2a • a–1 = (a + a–1)2 – 2a • a–1 = (a + 1/a)2 – 2 • 1 = 22 – 2 = 4 – 2 = 2. 4) якщо (1+a^2 )/a = 8, то a2 + a–2 = a2 + (a–1)2 + 2a • a–1 – 2a • a–1 = (a + a–1)2 – 2a • a–1 = (a + 1/a)2 – 2 • 1 = ((a^2+1)/a)2 – 2 = 82 – 2 = 64 – 2 = 62.