Find the least integral value of k for which the equation `x^(2)-2(k+2)x+12+k^(2)=0` has two different real roots.

Options

A. 0
B. 2
C. 3
D. 4

Answer

To find the least integral value of \( k \) for which the equation

\[ x^2 - 2(k+2)x + (12 + k^2) = 0 \]

has two different real roots, we need to analyze the discriminant of the quadratic equation.

### Step 1: Identify the coefficients
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -2(k + 2) \)
- \( c = 12 + k^2 \)

### Step 2: Write the discriminant condition
For the quadratic equation to have two different real roots, the discriminant \( D \) must be greater than zero:

\[ D = b^2 - 4ac > 0 \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = [-2(k + 2)]^2 - 4(1)(12 + k^2) > 0 \]

### Step 3: Simplify the discriminant
Calculating \( D \):

\[ D = 4(k + 2)^2 - 4(12 + k^2) \]

Factoring out the 4:

\[ D = 4\left[(k + 2)^2 - (12 + k^2)\right] > 0 \]

### Step 4: Expand and simplify the expression inside the brackets
Expanding \( (k + 2)^2 \):

\[ (k + 2)^2 = k^2 + 4k + 4 \]

Now substituting back into the discriminant:

\[ D = 4\left[k^2 + 4k + 4 - 12 - k^2\right] > 0 \]

This simplifies to:

\[ D = 4(4k - 8) > 0 \]

### Step 5: Solve the inequality
Dividing both sides by 4:

\[ 4k - 8 > 0 \]

Adding 8 to both sides:

\[ 4k > 8 \]

Dividing by 4:

\[ k > 2 \]

### Step 6: Find the least integral value of \( k \)
Since \( k \) must be greater than 2, the least integral value of \( k \) is:

\[ k = 3 \]

### Final Answer
Thus, the least integral value of \( k \) for which the equation has two different real roots is \( \boxed{3} \).

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