Question 37

1 + √(1 - k) is one of the solutions of the equation x ^ 2 - 2x + k = 0.

This solution arises from the quadratic formula applied to the equation, where the discriminant must be non-negative for real solutions.

A) (2 + √(4 + k))/2

This option does not satisfy the condition derived from the quadratic formula. When substituting this expression into the original equation, it does not yield a valid result in terms of satisfying the equation x^2 - 2x + k = 0.

B) 1 + √(1 - k)

This expression is indeed one of the solutions to the equation. Applying the quadratic formula gives the solutions as 1 ± √(1 - k), confirming that 1 + √(1 - k) is a valid solution when k is a real constant.

C) (- 2 + √(4 - k))/2

While this expression resembles a potential solution, it fails to satisfy the original quadratic equation as it does not meet the necessary conditions derived from the discriminant being non-negative. Thus, it is not a valid solution.

D) (2 + √(4 - k))/2

This option also does not fit as a solution to the equation. Substituting this back into the quadratic equation does not result in a valid equality, which means it cannot be considered a solution.

Conclusion

The selected correct answer, 1 + √(1 - k), is validated through the quadratic formula and meets all conditions for real roots. All other options either fail to satisfy the equation or do not derive from the proper form of the equation, confirming their incorrectness.