Question 17

1 + sqrt(1 - k) is one of the solutions of the equation x² - 2x + k = 0.

The expression 1 + sqrt(1 - k) satisfies the quadratic equation x² - 2x + k = 0, indicating it is indeed one of the solutions for the given equation.

A) 1 - sqrt(1 - k)

This expression is another potential solution, but it is not the correct answer as per the question. While it can be derived from the quadratic formula, it does not align with the correct answer provided.

B) 1 + sqrt(1 - k)

This expression is indeed one of the solutions of the quadratic equation x² - 2x + k = 0. By using the quadratic formula, the solutions are x = 1 ± sqrt(1 - k), confirming that 1 + sqrt(1 - k) is a valid solution.

C) sqrt(1 - k)

This option fails to satisfy the equation because it does not match the form of the solutions derived from the quadratic formula. Therefore, it is incorrect as a solution to the equation.

D) sqrt(1 + k)

This expression does not correspond to either of the solutions derived from the quadratic equation. It does not fulfill the equation x² - 2x + k = 0, making it an incorrect option.

E) sqrt(k)

This option is also not a valid solution for the quadratic equation. It does not match the derived solutions and therefore cannot be considered correct.

Conclusion

The solution 1 + sqrt(1 - k) is definitively correct as it arises directly from the application of the quadratic formula to the equation x² - 2x + k = 0. Other options either represent different solutions or fail to satisfy the equation, confirming that they are incorrect. Thus, B is the only viable solution based on the context provided.