Find the value of m so that the roots of the equation `(4-m)x^(2)+(2m+4)x+(8m+1)=0` may be equal.

Answer

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

\[ (4 - m)x^2 + (2m + 4)x + (8m + 1) = 0 \]

are equal, we need to use the condition that the discriminant \( D \) of a quadratic equation must be zero for the roots to be equal.

### Step 1: Identify coefficients
The general form of a quadratic equation is

\[ ax^2 + bx + c = 0 \]

From the given equation, we can identify:

- \( a = 4 - m \)
- \( b = 2m + 4 \)
- \( c = 8m + 1 \)

### Step 2: Write the discriminant
The discriminant \( D \) is given by:

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

### Step 3: Substitute the coefficients into the discriminant formula
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[ D = (2m + 4)^2 - 4(4 - m)(8m + 1) \]

### Step 4: Expand the discriminant
Now, we will expand both parts of the discriminant:

1. For \( (2m + 4)^2 \):

\[ (2m + 4)^2 = 4m^2 + 16m + 16 \]

2. For \( 4(4 - m)(8m + 1) \):

\[ 4(4 - m)(8m + 1) = 4[(32m - 8m^2) + (4 - m)] = 128m - 32m^2 + 16 - 4m \]

Putting it all together, we have:

\[ D = 4m^2 + 16m + 16 - (128m - 32m^2 + 16 - 4m) \]

### Step 5: Simplify the discriminant
Now, we simplify the expression:

\[ D = 4m^2 + 16m + 16 - 128m + 32m^2 - 16 + 4m \]

Combining like terms:

\[ D = (4m^2 + 32m^2) + (16m - 128m + 4m) + (16 - 16) \]

This simplifies to:

\[ D = 36m^2 - 108m \]

### Step 6: Set the discriminant to zero
For the roots to be equal, we set the discriminant \( D \) to zero:

\[ 36m^2 - 108m = 0 \]

### Step 7: Factor the equation
Factoring out the common term:

\[ 36m(m - 3) = 0 \]

### Step 8: Solve for \( m \)
Setting each factor to zero gives:

1. \( 36m = 0 \) → \( m = 0 \)
2. \( m - 3 = 0 \) → \( m = 3 \)

### Conclusion
Thus, the values of \( m \) for which the roots of the quadratic equation are equal are:

\[ m = 0 \quad \text{or} \quad m = 3 \]

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