Question

Найдите корни первообразной для функции

f(x)=x^2-4x+1 если один из них равен 2

Answer 1

 

$f(x)=x^2-4x+1\\ F(x) = int{x^2-4x+1}, dx = x^3/3 - 2x^2 + x + C\\ F(2) = 0, 2^3/3 - 2*2^2 + 2 + C = 0,\ 8/3 - 8 + 2 + C = 0, C = 10/3\\ F(x) = x^3/3 - 2x^2 + x + 10/3 =(x-2)(ax^2 + bx + c) =\ ax^3 + bx^2 + cx - 2ax^2 -2bx -2c =\ ax^3 + (b-2a)x^2 + (c-2b)x + (-2c)\\ a = 1/3,\ b-2/3 = -2, b = -4/3\ c+8/3 = 1, c = -5/3\\ F(x)= (x-2)(x^2/3 -(4/3)x - 5/3) = 0\\ (x^2/3 -(4/3)x - 5/3) = 0 | * 3\ x^2 - 4x - 5 = 0\\ x_1*x_2 = -5\ x_1+x_2 = 4\\ x_1 = -1, x_2 = 5\\ x_1 = 2, x_2 = -1, x_3 = 5$