Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation \( x^2 - 2bx + c^2 = 0 \) where \( b, c \) are positive real numbers. Let A be the arithmetic mean of \( \alpha \) and \( \beta \); and G be the geometric mean of \( \alpha \) and \( \beta \). What are the roots of the quadratic equation \( x^2 - (b + c)x + bc = 0 \) ?
- A. A, G ✓
- B. 2A, G
- C. A, 2G
- D. 2A, 2G
Correct Answer: A. A, G
Explanation
From the first equation, the sum of roots is 2b, so A = b. The product of roots is c^2, so G = c. The new equation has sum (b+c) and product bc, meaning its roots are b and c, which correspond to A and G.
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