Multiple choice

The equations $x^{2}+3x+5=0$ and $ax^{2}+bx+c=0$ have a common root. lf $a,\ b,\ c\in N$, then the least possible value of $a+b+c$ is equal to

  1. $3$
  2. $6$
  3. $9$
  4. $12$
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C Correct answer
Explanation

If x^2 + 3x + 5 = 0 and ax^2 + bx + c = 0 have a common root, the roots of the first are (-3 +/- sqrt(9-20))/2 = (-3 +/- i*sqrt(11))/2. Since the coefficients a, b, c are integers, the other root must be the conjugate. The quadratic is k(x^2 + 3x + 5) = 0. For a, b, c in N, the smallest k=1 gives a=1, b=3, c=5. Sum = 1+3+5 = 9.

AI explanation

The discriminant of x^2 + 3x + 5 = 0 is 3^2 - 4(1)(5) = -11, which is less than zero, so its roots are complex conjugates. For the equation ax^2 + bx + c = 0 with real coefficients to have a common root, it must share both complex roots, meaning the two equations are identical up to a constant factor. Therefore, a/1 = b/3 = c/5 = k for some positive integer k. To minimize a + b + c = k + 3k + 5k = 9k with natural numbers, we choose k = 1, yielding a = 1, b = 3, and c = 5, giving a minimum sum of 9.

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