Find the value of m if the equations `x^2+2x+3m=0` and `2x^2+3x+5m=0` have a common root

Options

A. (A) 15
B. (B) 8
C. (C) 2
D. (D) `-1`

Answer

To find the value of \( m \) such that the equations \( x^2 + 2x + 3m = 0 \) and \( 2x^2 + 3x + 5m = 0 \) have a common root, we can follow these steps:

### Step 1: Identify the coefficients
We have two quadratic equations:
1. \( x^2 + 2x + 3m = 0 \)
2. \( 2x^2 + 3x + 5m = 0 \)

From these equations, we can identify the coefficients:
- For the first equation, \( a_1 = 1 \), \( b_1 = 2 \), \( c_1 = 3m \)
- For the second equation, \( a_2 = 2 \), \( b_2 = 3 \), \( c_2 = 5m \)

### Step 2: Use the condition for common roots
If two quadratic equations have a common root, the following condition must hold:
\[ a_1 b_2 - a_2 b_1 = c_1 a_2 - c_2 a_1 \]
Substituting the values we identified:
\[ 1 \cdot 3 - 2 \cdot 2 = 3m \cdot 2 - 5m \cdot 1 \]

### Step 3: Simplify the equation
Now, simplify both sides:
\[ 3 - 4 = 6m - 5m \]
This simplifies to:
\[ -1 = m \]

### Step 4: Conclusion
Thus, the value of \( m \) is:
\[ m = -1 \]