If the equation `x^(2)-(2+m)x +(m^(2)-4m+4)=0` has equal roots then the values of m are

Options

A. (0,1)
B. `(2)/(3),6`
C. `(2)/(3),1`
D. 0,2

Answer

To solve the equation \( x^2 - (2 + m)x + (m^2 - 4m + 4) = 0 \) for the values of \( m \) that result in equal roots, we will follow these steps:

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

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant must be equal to zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero:
\[ b^2 - 4ac = 0 \]

### Step 3: Substitute the coefficients into the discriminant formula
Substituting the values of \( a \), \( b \), and \( c \):
\[ (-(2 + m))^2 - 4(1)(m^2 - 4m + 4) = 0 \]
This simplifies to:
\[ (2 + m)^2 - 4(m^2 - 4m + 4) = 0 \]

### Step 4: Expand and simplify the equation
Expanding \( (2 + m)^2 \):
\[ 4 + 4m + m^2 - 4(m^2 - 4m + 4) = 0 \]
Now, expand \( -4(m^2 - 4m + 4) \):
\[ -4m^2 + 16m - 16 \]
Combining these, we have:
\[ 4 + 4m + m^2 - 4m^2 + 16m - 16 = 0 \]
This simplifies to:
\[ -3m^2 + 20m - 12 = 0 \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ 3m^2 - 20m + 12 = 0 \]

### Step 6: Factor the quadratic equation
To factor \( 3m^2 - 20m + 12 = 0 \), 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 \( (m - 6) \):
\[ (3m - 2)(m - 6) = 0 \]

### Step 7: Solve for \( m \)
Setting each factor to zero gives:
1. \( 3m - 2 = 0 \) → \( m = \frac{2}{3} \)
2. \( m - 6 = 0 \) → \( m = 6 \)

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