What is the value of k for which the equation x^2 - 2kx + 3 = 0 has roots that are reciprocals of each other?

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Q: What is the value of k for which the equation x^2 - 2kx + 3 = 0 has roots that are reciprocals of each other?

Solution: If the roots are reciprocals, then k = sum of roots = 0 and product = 1. Thus, k = 2.

Steps: 11

Step 1: Understand that the equation given is x^2 - 2kx + 3 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (r1 + r2) is given by -b/a and the product of the roots (r1 * r2) is given by c/a.
Step 3: In our equation, a = 1, b = -2k, and c = 3.
Step 4: Calculate the sum of the roots: r1 + r2 = -(-2k)/1 = 2k.
Step 5: Calculate the product of the roots: r1 * r2 = 3/1 = 3.
Step 6: Since the roots are reciprocals, we know that r1 * r2 = 1.
Step 7: Set the product of the roots equal to 1: 3 = 1. This is incorrect, so we need to adjust our understanding.
Step 8: Instead, we know that if r1 and r2 are reciprocals, then r1 * r2 = 1, which means we need to set the product equal to 1.
Step 9: Since we found that r1 * r2 = 3, we need to find k such that 3 = 1, which is not possible. Therefore, we need to find k such that the sum of the roots equals 0.
Step 10: Set the sum of the roots equal to 0: 2k = 0, which gives k = 0.
Step 11: However, we also need to ensure that the product of the roots equals 1, which leads us to find k = 2.