What is the radix of the numbers if the solution to the quadratic equation…
2020
What is the radix of the numbers if the solution to the quadratic equation \(x^2-10x+26=0\) is \(𝑥=4\) and \(𝑥=7\)?
- A.
8
- B.
9
- C.
10
- D.
11
Attempted by 17 students.
Show answer & explanation
Correct answer: D
Key idea: Interpret the written coefficients in an unknown base r.
The symbol "10" in base r represents r in decimal, and "26" in base r represents 2r + 6. So the equation becomes x^2 - r x + (2r + 6) = 0.
Use Vieta's formulas:
Sum of roots = r. Given roots 4 and 7, 4 + 7 = 11, so r = 11.
Product of roots = 2r + 6. With r = 11, 2×11 + 6 = 28, and 4×7 = 28, so the product check matches.
Also check digit validity: the digits used in the coefficients are 0, 1, 2, and 6, all of which are less than 11, so base 11 is valid.
Conclusion: The radix is 11.
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