How to use the idea of quadratic equation to find the minimum value of function y = (x ^ 2-2x + 6) / (x + 1), (x ≥ 0) We all know the solution of mean value inequality and derivative, but it is mentioned in the book that the minimum value of the function can also be obtained by using the discriminant of quadratic equation. How to solve it by using quadratic equation?

How to use the idea of quadratic equation to find the minimum value of function y = (x ^ 2-2x + 6) / (x + 1), (x ≥ 0) We all know the solution of mean value inequality and derivative, but it is mentioned in the book that the minimum value of the function can also be obtained by using the discriminant of quadratic equation. How to solve it by using quadratic equation?

The solution of mean inequality and derivative is not mentioned
Y = (x ^ 2-2x + 6) / (x + 1) = ((x-1) ^ 2 + 5) / (x + 1). When x ≥ 0, the function value y is greater than 0
And: x ^ 2-2x + 6 = YX + y, that is: x ^ 2 - (y + 2) x + 6-y = 0, the discriminant △≥ 0 is meaningful
That is: (y + 2) ^ 2-4 (6-y) = y ^ 2 + 8y-20 = (Y-2) (y + 10) ≥ 0, i.e. y ≥ 2 or Y ≤ - 10
So: y ≥ 2, that is: the minimum value of Y is 2