Consider the quadratic equation \(x^2-13x+36=0\) with coefficients in a base…
2017
Consider the quadratic equation \(x^2-13x+36=0\) with coefficients in a base \(b\). The solutions of this equation in the same base \(b\) are \(x\) = 5 and \(x\) = 6. Then \(b\) = _____
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Correct answer: 8
Answer: 8
Interpret the numerals in base b: 13 in base b equals b + 3, and 36 in base b equals 3b + 6.
Use Vieta's formulas for x^2 - 13x + 36 = 0: the sum of the roots equals 13 in base b, and the product equals 36 in base b.
Sum: 5 + 6 = 11, so b + 3 = 11, which gives b = 8.
Product check: 5 × 6 = 30, and 3b + 6 with b = 8 equals 24 + 6 = 30, consistent.
Digit constraint: all digits used (1, 3, 3, 6, 5, 6) are less than 8, so base 8 is valid.
Thus the base is 8.
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