Пусть x1, x2 - корни уравнения 2x^2+3x-4=0. Чему равно значение выражения 16 (x1^4+x2^4) ?

По теореме Виета

x1 + x2 = - b/a = - 3/2

x1*x2 = c/a = - 4/2 = - 2

x1^4 + x2^4 = x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2 =

= (x1^2 + x2^2) ^2 - 2 (x1*x2) ^2 =

= (x1^2 + 2x1*x2 + x2^2 - 2x1*x2) ^2 - 2 (x1*x2) ^2 =

= ((x1 + x2) ^2 - 2x1*x2) ^2 - 2 (x1*x2) ^2 = ((-3/2) ^2 - 2 (-2)) ^2 - 2 (-2) ^2 =

= (9/4 + 4) ^2 - 2*4 = (25/4) ^2 - 8 = 625/16 - 128/16 = 497/16

16 (x1^4+x2^4) = 16*497/16 = 497