If the roots of $x^2-\mathit{bx}+c=0$are each decreased by $2$, then resulting equation is $x^2-2x+1=0$, if and only if

Complete Solution:

From the equation:

x² - 2x + 1 = 0,

The roots are:

x = 1

(a double root, as it factors as (x - 1)² = 0).

If the roots of the original equation:

x² - bx + c = 0

are decreased by 2 to become 1, then the roots of the original equation must have been:

1 + 2 = 3

and

1 + 2 = 3.

Thus, the roots of the original equation are 3 and 3 (a repeated root).

For the original equation:

x² - bx + c = 0,

The sum and product of the roots are related to the coefficients:

Sum of the roots:

3 + 3 = 6 = b

Product of the roots:

3 × 3 = 9 = c

The transformation holds if and only if:

b = 6 and c = 9.

Final Answer:

The resulting equation: x² - 2x + 1 = 0 holds if and only if: b = 6 and c = 9.