If `alpha,beta,gamma in R, alpha+beta+gamma=4 " and " alpha^(2)+beta^(2)+gamma^(2)=6`, the number of integers lie in the exhaustive range of `alpha` is ……… .

Answer

To solve the problem, we need to analyze the given equations:

1. \( \alpha + \beta + \gamma = 4 \)
2. \( \alpha^2 + \beta^2 + \gamma^2 = 6 \)

We want to find the number of integers that can take the value of \( \alpha \).

### Step 1: Use the identity for the sum of squares
We can use the identity that relates the sum of squares to the square of the sum:

\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]

Substituting the first equation into this identity:

\[ 6 = 4^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]

This simplifies to:

\[ 6 = 16 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]

### Step 2: Rearranging the equation
Rearranging gives us:

\[ 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 16 - 6 \]

\[ 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 10 \]

\[ \alpha\beta + \beta\gamma + \gamma\alpha = 5 \]

### Step 3: Form a quadratic equation
Now we can think of \( \alpha, \beta, \gamma \) as the roots of a polynomial. Let \( \alpha, \beta, \gamma \) be the roots of the polynomial:

\[ x^3 - sx^2 + px - r = 0 \]

where \( s = \alpha + \beta + \gamma = 4 \), \( p = \alpha\beta + \beta\gamma + \gamma\alpha = 5 \), and \( r = \alpha\beta\gamma \) (which we do not need to find).

### Step 4: Finding the range of \( \alpha \)
Now we need to find the possible values of \( \alpha \). Since \( \alpha + \beta + \gamma = 4 \), we can express \( \beta + \gamma \) as:

\[ \beta + \gamma = 4 - \alpha \]

And from the second equation:

\[ \beta^2 + \gamma^2 = 6 - \alpha^2 \]

Using the identity \( \beta^2 + \gamma^2 = (\beta + \gamma)^2 - 2\beta\gamma \):

\[ 6 - \alpha^2 = (4 - \alpha)^2 - 2\beta\gamma \]

### Step 5: Finding bounds for \( \alpha \)
We can analyze the values of \( \alpha \):

1. If \( \alpha = 2 \), then \( \beta + \gamma = 2 \) and \( \beta^2 + \gamma^2 = 2 \), which gives \( \beta = 1, \gamma = 1 \).
2. If \( \alpha = 1 \), then \( \beta + \gamma = 3 \) and \( \beta^2 + \gamma^2 = 5 \). Possible pairs are \( (2, 1) \) or \( (1, 2) \).
3. If \( \alpha = 0 \), then \( \beta + \gamma = 4 \) and \( \beta^2 + \gamma^2 = 6 \). Possible pairs are \( (2, 2) \).
4. If \( \alpha = -1 \), then \( \beta + \gamma = 5 \) and \( \beta^2 + \gamma^2 = 7 \).
5. If \( \alpha = -2 \), then \( \beta + \gamma = 6 \) and \( \beta^2 + \gamma^2 = 10 \).

### Step 6: Checking integer values
The possible integer values for \( \alpha \) that satisfy both conditions are \( -2, -1, 0, 1, 2 \).

Thus, the integers that \( \alpha \) can take are:

- \( -2 \)
- \( -1 \)
- \( 0 \)
- \( 1 \)
- \( 2 \)

### Conclusion
The number of integers in the exhaustive range of \( \alpha \) is **5**.

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