Multiple choice

If the equations $x^2+4x+5=0$ and $ax^2+bx+c=0$ have a common root (where $a,b,c \in N$), then the least value of $a+b+c$ is equal to:

  1. $10$
  2. $15$
  3. $20$
  4. $25$
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A Correct answer
Explanation

x^2+4x+5=0 has roots -2 +/- i. If ax^2+bx+c=0 has a common root, it must have both roots (since coefficients are real). So ax^2+bx+c = k(x^2+4x+5). For a,b,c to be in N, k must be a positive integer. Min value is k=1, giving a=1, b=4, c=5. Sum = 1+4+5 = 10.

AI explanation

The discriminant of x^2 + 4x + 5 = 0 is 16 - 20 = -4, which is less than zero, so the roots are complex. If a second quadratic equation with real coefficients has a common root with this equation, it must share both complex conjugate roots. This means the two equations must be identical up to a constant multiple, so ax^2 + bx + c = 0 must equal k(x^2 + 4x + 5). Using the smallest possible positive integer k = 1 gives a = 1, b = 4, and c = 5. The least value of a + b + c is therefore 1 + 4 + 5 = 10.

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