Multiple choice

For $a,\ b,\ c\ \in\ R$ and $a\ \neq 0$, the equation $ax^{2}+bx+c=0$ and $x^{2}+2x+3=0$ have atlest one common root if $\dfrac{a}{\lambda}=\dfrac{b}{\mu}=\dfrac{c}{\Psi }$ are positive integers. The least value of $(\lambda+\mu+\Psi )$ is

  1. $6$
  2. $5$
  3. $7$
  4. $4$
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A Correct answer
Explanation

For common roots, the ratio of coefficients must be equal. x^2+2x+3=0 has roots -1 +/- i*sqrt(2). For ax^2+bx+c=0 to share a root, a/1 = b/2 = c/3 = k. Since a,b,c are positive integers, the smallest k=1 gives a=1, b=2, c=3. Sum = 1+2+3 = 6.

AI explanation

The equation x^2 + 2x + 3 = 0 has a discriminant of 4 - 12 = -8, which is less than zero. This means the roots are complex, so any other quadratic equation with real coefficients that shares a common root must actually share both complex conjugate roots. Therefore, the equations must be identical up to a constant multiple, meaning a/1 = b/2 = c/3. For a, b, and c to be positive integers, the smallest possible multiplier is 1, which makes lambda = 1, mu = 2, and Psi = 3. The least value of lambda + mu + Psi is 1 + 2 + 3 = 6.

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