For what value of m equation `4x^(2) + 6mx + 9=0` have equal roots

Answer

To find the value of \( m \) for which the equation \( 4x^2 + 6mx + 9 = 0 \) has equal roots, we need to use the concept of the discriminant. The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are equal when the discriminant \( D \) is equal to zero. The discriminant is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify coefficients
In our equation \( 4x^2 + 6mx + 9 = 0 \):
- \( a = 4 \)
- \( b = 6m \)
- \( c = 9 \)

### Step 2: Set up the discriminant
We set the discriminant to zero for equal roots:

\[ D = (6m)^2 - 4 \cdot 4 \cdot 9 = 0 \]

### Step 3: Calculate the discriminant
Now, we calculate \( D \):

\[ (6m)^2 = 36m^2 \]
\[ 4 \cdot 4 \cdot 9 = 144 \]

So, we have:

\[ 36m^2 - 144 = 0 \]

### Step 4: Solve for \( m^2 \)
Now we can solve for \( m^2 \):

\[ 36m^2 = 144 \]
\[ m^2 = \frac{144}{36} \]
\[ m^2 = 4 \]

### Step 5: Find \( m \)
Taking the square root of both sides gives us:

\[ m = \pm 2 \]

### Conclusion
Thus, the values of \( m \) for which the equation \( 4x^2 + 6mx + 9 = 0 \) has equal roots are:

\[ m = 2 \quad \text{or} \quad m = -2 \]

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