If equations `ax^(2)-bx+c=0` (where `a,b,c epsilonR` and `a!=0`) and `x^(2)+2x+3=0` have a common root, then show that `a:b:c=1:2:3`

Explanation

To solve the problem, we need to show that if the equations \( ax^2 - bx + c = 0 \) and \( x^2 + 2x + 3 = 0 \) have a common root, then the ratio \( a:b:c = 1:2:3 \).

### Step 1: Identify the common root
Let \( r \) be the common root of both equations. Then, substituting \( r \) into both equations gives us:
1. \( ar^2 - br + c = 0 \) (from the first equation)
2. \( r^2 + 2r + 3 = 0 \) (from the second equation)

### Step 2: Solve the second equation for its roots
...