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

Rationale

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

To solve the quadratic equation x^2 - 2x + k = 0, we can apply the quadratic formula x = [2 ± √(4 - 4k)] / 2. This simplifies to x = 1 ± √(1 - k), revealing that one of the solutions is indeed 1 + √(1 - k).

A ((2 + √(4 + k))/2) YOUR ANSWER

This choice introduces an incorrect term, √(4 + k), which does not align with the derivation from the quadratic formula. The term under the square root should reflect the discriminant of the equation, specifically (4 - 4k), not (4 + k). As such, this expression does not yield a valid solution for the equation provided.

B (1 + √(1 - k)) CORRECT

This choice is derived directly from the simplification of the quadratic formula applied to the equation. By substituting the values from the standard form, this expression represents one of the two possible solutions of the quadratic, confirming its validity as a solution.

C ((- 2 + √(4 - k))/2) YOUR ANSWER

While this expression is a rearrangement of the quadratic formula, it corresponds to the other solution, 1 - √(1 - k), rather than the one indicated in the question. Thus, it does not meet the criteria of being one of the solutions specified.

D ((2 + √(4 - k))/2) YOUR ANSWER

This expression simplifies to 1 + √(1 - k), similar to choice B but is formatted incorrectly in relation to the original equation. It does not represent a solution derived from the standard quadratic formula as presented and, therefore, is not the correct answer.

Conclusion

In summary, the correct solution to the equation x^2 - 2x + k = 0 is 1 + √(1 - k), as derived from the quadratic formula. The other choices either misrepresent the solutions or contain inaccuracies in their formulation relative to the equation. Understanding these relationships is crucial for solving quadratic equations and applying the quadratic formula correctly.

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

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

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