Multiple choice

If equations $ax^2+bx+c=0, (a, b, c \in R, a\neq 0)$ and $2x^2+3x+4=0$ have a common root, then $a : b : c$ equals :

  1. $1 : 2 : 3$
  2. $2 : 3 : 4$
  3. $4 : 3 : 2$
  4. $3 : 2 : 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If ax^2 + bx + c = 0 and 2x^2 + 3x + 4 = 0 have a common root, the coefficients must be proportional if the roots are identical, or satisfy the resultant condition. Comparing the equations, if they share the same roots, a/2 = b/3 = c/4, so a:b:c = 2:3:4.

AI explanation

Using the discriminant of 2x^2 + 3x + 4 = 0, we get 3^2 - 4(2)(4) = -23, which is less than zero, meaning this equation has complex roots. If ax^2 + bx + c = 0 has a common root with it, both equations must share both roots because complex roots for equations with real coefficients occur in conjugate pairs. Therefore, the ratios of their corresponding coefficients must be equal, giving a/2 = b/3 = c/4. This makes the ratio a : b : c equal to 2 : 3 : 4.

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