Which of the following is one of the solutions of the equation x² - 2x + k = 0, where k is a real constant?

Rationale

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

To find the solutions of the quadratic equation x² - 2x + k = 0, we can use the quadratic formula, which states that the solutions are given by x = [2 ± sqrt(4 - 4k)]/2. This simplifies to x = 1 ± sqrt(1 - k), indicating that one of the solutions is indeed 1 + sqrt(1 - k).

A (1 - sqrt(1 - k)) YOUR ANSWER

This is incorrect because the actual solutions come from x= 1+- sqrt(1-k) so the expression under the square root must be 1−k, not k. Therefore, 1+- sqrt(k) does not satisfy the quadratic equation in general and is not a valid solution.

B (1 + sqrt(1 - k)) CORRECT

This option corresponds directly to one of the solutions derived from the quadratic formula applied to the equation x² - 2x + k = 0. It represents a valid solution that can exist for various real values of k, confirming its correctness.

C (sqrt(1 - k)) YOUR ANSWER

While sqrt(1 - k) could represent a potential value derived from the equation, it is not one of the solutions of the given quadratic. The solutions are specifically 1 ± sqrt(1 - k), and this choice omits the necessary addition of 1, thus failing to satisfy the equation.

D (sqrt(1 + k)) YOUR ANSWER

This choice does not relate to the original equation's solutions. The expression sqrt(1 + k) does not appear in the process of solving x² - 2x + k = 0 and thus does not represent any solution derived from it.

E (sqrt(k)) YOUR ANSWER

Similar to option D, sqrt(k) does not correspond to any solution of the quadratic equation. The roots found using the quadratic formula do not involve k alone in this manner, making this choice irrelevant.

Conclusion

The quadratic equation x² - 2x + k = 0 has solutions that can be derived using the quadratic formula, leading to the conclusions that include 1 + sqrt(1 - k) as one of the roots. The other choices either fail to represent valid solutions or do not connect with the equation's structure, underscoring the significance of understanding how to apply the quadratic formula to find accurate results.

Which of the following is one of the solutions of the equation x² - 2x + k = 0, where k is a real constant?

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

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