The complete set of values of `k ,` for which the quadratic `x^2-k x+k+2=0` has equal roots, consists of

Options

A. `2+sqrt(12)`
B. `2+-sqrt(12)`
C. `2-sqrt(12)`
D. `-2-sqrt(12)`

Answer

To find the complete set of values of \( k \) for which the quadratic equation \( x^2 - kx + (k + 2) = 0 \) has equal roots, we need to 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:
- \( a = 1 \)
- \( b = -k \)
- \( c = k + 2 \)

### Step 2: Use the condition for equal roots
A quadratic equation has equal roots if its discriminant is zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero for our equation:
\[ D = (-k)^2 - 4(1)(k + 2) = 0 \]

### Step 3: Simplify the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant:
\[ k^2 - 4(k + 2) = 0 \]
Expanding this:
\[ k^2 - 4k - 8 = 0 \]

### Step 4: Solve the quadratic equation
Now, we will solve the quadratic equation \( k^2 - 4k - 8 = 0 \) using the quadratic formula:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -4 \), and \( c = -8 \):
\[ k = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-8)}}{2(1)} \]
Calculating the discriminant:
\[ k = \frac{4 \pm \sqrt{16 + 32}}{2} \]
\[ k = \frac{4 \pm \sqrt{48}}{2} \]

### Step 5: Simplify further
We can simplify \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \]
Substituting back:
\[ k = \frac{4 \pm 4\sqrt{3}}{2} \]
\[ k = 2 \pm 2\sqrt{3} \]

### Step 6: Final answer
Thus, the complete set of values of \( k \) for which the quadratic equation has equal roots is:
\[ k = 2 + 2\sqrt{3} \quad \text{and} \quad k = 2 - 2\sqrt{3} \]