Let `alpha, beta` be real roots of the quadratic equatin `x ^(2) +kx+ (k^(2) +2k -4)=0,` then the maximum value of `(alpha ^(2) +beta ^(2)) ` is equal to :

Options

A. 9
B. 10
C. 11
D. 12

Answer

To solve the problem, we need to find the maximum value of \( \alpha^2 + \beta^2 \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + kx + (k^2 + 2k - 4) = 0 \).

### Step 1: Identify the coefficients
The given quadratic equation is:
\[ x^2 + kx + (k^2 + 2k - 4) = 0 \]
Here, we can identify:
- \( a = 1 \)
- \( b = k \)
- \( c = k^2 + 2k - 4 \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -k \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = k^2 + 2k - 4 \)

### Step 3: Express \( \alpha^2 + \beta^2 \) in terms of \( \alpha + \beta \) and \( \alpha \beta \)
We know the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (-k)^2 - 2(k^2 + 2k - 4) \]
This simplifies to:
\[ \alpha^2 + \beta^2 = k^2 - 2(k^2 + 2k - 4) \]

### Step 4: Simplify the expression
Expanding the expression:
\[ \alpha^2 + \beta^2 = k^2 - 2k^2 - 4k + 8 \]
Combining like terms:
\[ \alpha^2 + \beta^2 = -k^2 - 4k + 8 \]

### Step 5: Find the maximum value
To find the maximum value of \( -k^2 - 4k + 8 \), we can rewrite it in a standard quadratic form:
\[ \alpha^2 + \beta^2 = - (k^2 + 4k - 8) \]
The expression \( k^2 + 4k - 8 \) is a quadratic function that opens upwards (since the coefficient of \( k^2 \) is positive). The maximum value of \( - (k^2 + 4k - 8) \) occurs at the vertex of the parabola.

### Step 6: Find the vertex
The vertex \( k \) of the quadratic \( k^2 + 4k - 8 \) is given by:
\[ k = -\frac{b}{2a} = -\frac{4}{2 \cdot 1} = -2 \]

### Step 7: Substitute \( k = -2 \) back into the expression
Now substituting \( k = -2 \) into the expression for \( \alpha^2 + \beta^2 \):
\[ \alpha^2 + \beta^2 = -((-2)^2 + 4(-2) - 8) \]
Calculating:
\[ = - (4 - 8 - 8) = -(-12) = 12 \]

### Conclusion
Thus, the maximum value of \( \alpha^2 + \beta^2 \) is:
\[ \boxed{12} \]