For what values of m the equation `(1 + m)x^2 - 2(1 + 3m)x + (1 + 8m) = 0` has equal roots.

Answer

To find the values of \( m \) for which the equation

\[ (1 + m)x^2 - 2(1 + 3m)x + (1 + 8m) = 0 \]

has equal roots, we need to set the discriminant of the quadratic equation to zero. The discriminant \( D \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

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

### Step 1: Identify coefficients \( a \), \( b \), and \( c \)

From the given equation, we can identify:
- \( a = 1 + m \)
- \( b = -2(1 + 3m) \)
- \( c = 1 + 8m \)

### Step 2: Write the expression for the discriminant

Now, substituting these values into the discriminant formula:

\[ D = \left(-2(1 + 3m)\right)^2 - 4(1 + m)(1 + 8m) \]

### Step 3: Simplify the discriminant

Calculating \( D \):

1. Calculate \( b^2 \):
\[ b^2 = \left(-2(1 + 3m)\right)^2 = 4(1 + 3m)^2 = 4(1 + 6m + 9m^2) = 4 + 24m + 36m^2 \]

2. Calculate \( 4ac \):
\[ 4ac = 4(1 + m)(1 + 8m) = 4(1 + 8m + m + 8m^2) = 4(1 + 9m + 8m^2) = 4 + 36m + 32m^2 \]

3. Substitute back into the discriminant:
\[ D = (4 + 24m + 36m^2) - (4 + 36m + 32m^2) \]

4. Simplify:
\[ D = 4 + 24m + 36m^2 - 4 - 36m - 32m^2 = (36m^2 - 32m^2) + (24m - 36m) + 0 = 4m^2 - 12m \]

### Step 4: Set the discriminant to zero

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

\[ 4m^2 - 12m = 0 \]

### Step 5: Factor the equation

Factoring out \( 4m \):

\[ 4m(m - 3) = 0 \]

### Step 6: Solve for \( m \)

Setting each factor to zero gives us:

1. \( 4m = 0 \) → \( m = 0 \)
2. \( m - 3 = 0 \) → \( m = 3 \)

### Final Answer

The values of \( m \) for which the equation has equal roots are:

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

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