Multiple choice

If $a,b,c$ belong to R and equations $ax^2+bx+c=0$ and $x^2+2x+9=0$ have a common root, find $a:b:c$.

  1. $1:3:4$
  2. $1:2:9$
  3. $2:5:7$

  4. None of these
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B Correct answer
Explanation

If the equations have a common root, the coefficients must be proportional. Comparing ax^2 + bx + c = 0 with x^2 + 2x + 9 = 0, the ratio a:b:c must be 1:2:9.

AI explanation

We first determine the roots of the equation x squared plus 2x plus 9 equals 0 using the quadratic formula, which gives x equals -1 plus or minus 2i times the square root of 2. Since these are complex roots and the coefficients a, b, and c are real numbers, both complex roots must be shared by the two equations. Therefore, the second equation must simply be a direct scalar multiple of the first equation. This means the ratio a to b to c must be identical to the ratio of the coefficients in the original equation, which is 1 to 2 to 9.

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