Question

Докажите, что многочлен х^2-2х+у^2-4у+6 при любых значениях входящих в него переменных принимает положительные значения.

Answer 1

(x²-2x+1)+(y²-4y+4)+1=(x-1)²+(y-2)²+1
Квадрат-число положительное,сумма положительных больше 0

Answer 2

Checking the Polynomial for Positive Values

To prove that the polynomial x^2 - 2x + y^2 - 4y + 6 takes positive values for any input values of its variables, we can analyze its components.

The given polynomial can be expressed as the sum of squares and a constant term: (x^2 - 2x + 1) + (y^2 - 4y + 4) + 1.

This can be further simplified to: (x - 1)^2 + (y - 2)^2 + 1.

Conclusion

The polynomial can be rewritten as the sum of squares of two terms, each of which is always non-negative, plus a constant term. Therefore, the given polynomial always takes positive values for any input values of its variables.

[[SOURCE #1]]