Вправи 1001 - 1100 » 1054

1054. Не розв’язуючи рівняння х2 – 2х – 9 = 0, обчисли. x² – 2x – 9 = 0; а) x1²x2 + x1x2²; x1 + x2 = 2; x1 • x2 = –9. x1²x2 + x1x2² = x1x2(x1 + x2) = –9 • 2 = –18. б) x1² + x2²; x1 + x2 = 2; x1 • x2 = –9. x1² + x22 = (x1 + x2)2 – 2x1x2 = 22 – 2 • (–9) = 4 + 18 = 22. в) (x_1^2)/x_2 + (x_2^2)/x_1 ; x1 + x2 = 2; x1 • x2 = –9. (x_1^2)/x_2 + (x_2^2)/x_1 = (x_1^3+ x_2^3 )/(x_1 x_2 ) = ((x_1+ x_2 )(x_1^2- x_1 x_2+ x_2^2))/(x_1 x_2 ) = (2 •((x_1+ x_2 )^2- 2x_1 x_2- x_1 x_2))/(-9) = (2 •(2^2- 3x_1 x_2))/(-9) = (2 •(4-3 •(-9)))/(-9) = (2 •(4+27))/(-9) = (2 •31)/(-9) = 63/(-9) = –7.