If the equation `x^2-(2 + m)x + (m2-4m +4) = 0` has coincident roots, then

Options

A. m = 0, 1
B. m = 2/3, 1
C. m = 0, 2
D. m = 2/3, 6

Answer

To solve the problem, we need to determine the values of \( m \) for which the quadratic equation

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

has coincident roots. This occurs when the discriminant of the quadratic equation is equal to zero.

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = -(2 + m) \)
- \( c = m^2 - 4m + 4 \)

### Step 2: Write the discriminant
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
For our equation, substituting the values of \( a \), \( b \), and \( c \):
\[ D = [-(2 + m)]^2 - 4 \cdot 1 \cdot (m^2 - 4m + 4) \]

### Step 3: Simplify the discriminant
Calculating \( D \):
\[ D = (2 + m)^2 - 4(m^2 - 4m + 4) \]
Expanding \( (2 + m)^2 \):
\[ D = (4 + 4m + m^2) - 4(m^2 - 4m + 4) \]
Distributing the \( -4 \):
\[ D = 4 + 4m + m^2 - 4m^2 + 16m - 16 \]
Combining like terms:
\[ D = -3m^2 + 20m - 12 \]

### Step 4: Set the discriminant to zero
For the roots to be coincident, we set the discriminant equal to zero:
\[ -3m^2 + 20m - 12 = 0 \]
Multiplying through by -1 to simplify:
\[ 3m^2 - 20m + 12 = 0 \]

### Step 5: Factor the quadratic equation
To factor \( 3m^2 - 20m + 12 \), we look for two numbers that multiply to \( 3 \times 12 = 36 \) and add to \( -20 \). The numbers are \( -18 \) and \( -2 \):
\[ 3m^2 - 18m - 2m + 12 = 0 \]
Grouping the terms:
\[ 3m(m - 6) - 2(m - 6) = 0 \]
Factoring out \( (m - 6) \):
\[ (m - 6)(3m - 2) = 0 \]

### Step 6: Solve for \( m \)
Setting each factor to zero gives:
1. \( m - 6 = 0 \) → \( m = 6 \)
2. \( 3m - 2 = 0 \) → \( m = \frac{2}{3} \)

### Final Answer
The values of \( m \) for which the quadratic equation has coincident roots are:
\[ m = 6 \quad \text{and} \quad m = \frac{2}{3} \]