Не вычисляя корней уравнения 2х²-6х+3=0 Найдите а) [latex] x1 x2^{5} + x1^{5} x2[/latex] б) [latex] x1^{4} + x2^{4} [/latex] в)[latex] \frac{x1}{ x2^{2} } + \frac{x2}{ x1^{2} } [/latex]

По теореме Виета x1 + x2 = 6/2 = 3; x1*x2 = 3/2 = 1,5. a) x1*x2^5 + x1^5*x2 = x1*x2*(x1^4 + x2^4) = = 1,5*(x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2) = = 1,5*((x1^2 + x2^2)^2 - 2*(1,5)^2) = = 1,5*((x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2) - 2*2,25) = = 1,5*( [ (x1+x2)^2 - 2*1,5 ]^2 - 4,5) = 1,5*((3^2 - 3)^2 - 4,5) = = 1,5*(6^2 - 4,5) = 1,5*(36 - 4,5) = 1,5*31,5 = 47,25 b) x1^4 + x2^4 = x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2 = (x1^2 + x2^2)^2 - 2*(1,5)^2 = (x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2) - 2*2,25 = [ (x1+x2)^2 - 2*1,5 ]^2 - 4,5 = (3^2 - 3)^2 - 4,5 = 36 - 4,5 = 31,5 c) [latex] \frac{x1}{x2^2} + \frac{x2}{x1^2} = \frac{x1^3+x2^3}{x1^2*x2^2} = \frac{(x1+x2)(x1^2-x1*x2+x2^2)}{(x1*x2)^2} =[/latex] [latex]= \frac{3(x1^2+2x1x2+x2^2-3x1x2)}{(1,5)^2} = \frac{1,5*2((x1+x2)^2-3*1,5)}{(1,5)^2} = \frac{2(3^2-4,5)}{1,5} = \frac{2*4,5}{1,5}=6 [/latex]