If `(m^2-3)x^2+3mx+ 3m+1=0` has roots which are reciprocals of each other, then the value of m equals to

Options

A. 4
B. 1
C. 2
D. none of these

Answer

To find the value of \( m \) such that the quadratic equation

\[ (m^2 - 3)x^2 + 3mx + (3m + 1) = 0 \]

has roots that are reciprocals of each other, we can follow these steps:

### Step 1: Understand the condition for reciprocal roots
For a quadratic equation \( ax^2 + bx + c = 0 \), if the roots are reciprocals of each other, then the product of the roots \( \alpha \times \beta = 1 \). According to Vieta's formulas, the product of the roots can also be expressed as \( \frac{c}{a} \).

### Step 2: Identify coefficients
From the given quadratic equation, we have:
- \( a = m^2 - 3 \)
- \( b = 3m \)
- \( c = 3m + 1 \)

### Step 3: Set up the equation using the product of roots
Since the product of the roots is 1, we can set up the equation:

\[ \frac{c}{a} = 1 \]

Substituting the values of \( c \) and \( a \):

\[ \frac{3m + 1}{m^2 - 3} = 1 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 3m + 1 = m^2 - 3 \]

### Step 5: Rearrange the equation
Rearranging the equation results in:

\[ m^2 - 3m - 4 = 0 \]

### Step 6: Factor the quadratic equation
To solve \( m^2 - 3m - 4 = 0 \), we can factor it:

\[ (m - 4)(m + 1) = 0 \]

### Step 7: Solve for \( m \)
Setting each factor to zero gives us:

\[ m - 4 = 0 \quad \Rightarrow \quad m = 4 \]

\[ m + 1 = 0 \quad \Rightarrow \quad m = -1 \]

### Conclusion
Thus, the values of \( m \) are:

\[ m = 4 \quad \text{and} \quad m = -1 \]