Find the value of k for which the following equation has equal roots :
`x^(2)+4kx+(k^(2)-k+2)=0`

Options

A. `-5 or 2/(3)`
B. `-1 or 2/(3)`
C. `0 or 2/(3)`
D. None

Answer

To find the value of \( k \) for which the quadratic equation \( x^2 + 4kx + (k^2 - k + 2) = 0 \) has equal roots, we will follow these steps:

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

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant \( D \) must be equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (4k)^2 - 4(1)(k^2 - k + 2) \]

### Step 3: Simplify the discriminant
Calculating \( D \):
\[ D = 16k^2 - 4(k^2 - k + 2) \]
Distributing the \( -4 \):
\[ D = 16k^2 - 4k^2 + 4k - 8 \]
Combining like terms:
\[ D = 12k^2 + 4k - 8 \]

### Step 4: Set the discriminant to zero
For equal roots, we set the discriminant \( D \) equal to zero:
\[ 12k^2 + 4k - 8 = 0 \]

### Step 5: Simplify the equation
We can divide the entire equation by 4 to simplify:
\[ 3k^2 + k - 2 = 0 \]

### Step 6: Factor the quadratic equation
Now, we will factor \( 3k^2 + k - 2 \):
\[ 3k^2 + 3k - 2k - 2 = 0 \]
Grouping the terms:
\[ (3k^2 + 3k) + (-2k - 2) = 0 \]
Factoring by grouping:
\[ 3k(k + 1) - 2(k + 1) = 0 \]
Factoring out \( (k + 1) \):
\[ (3k - 2)(k + 1) = 0 \]

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

### Conclusion
The values of \( k \) for which the equation has equal roots are:
\[ k = \frac{2}{3} \quad \text{and} \quad k = -1 \]

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