If `1 in (alpha, beta)` where `alpha, beta` are the roots of the equation `x^(2)-a(x+1)+3=0`, then

Options

A. `a gt 2`
B. `a lt -6 and a gt 2`
C. `-6 lt a lt 2`
D. `a lt 2`

Answer

To solve the problem, we need to analyze the quadratic equation given and the condition that the number 1 lies between its roots, alpha and beta.

### Step 1: Rewrite the quadratic equation
The given quadratic equation is:
\[ x^2 - a(x + 1) + 3 = 0 \]
We can rewrite it as:
\[ x^2 - ax - a + 3 = 0 \]
This can be simplified to:
\[ x^2 - ax + (3 - a) = 0 \]

### Step 2: Identify coefficients
From the standard form of a quadratic equation \(Ax^2 + Bx + C = 0\), we identify:
- \(A = 1\)
- \(B = -a\)
- \(C = 3 - a\)

### Step 3: Use the quadratic formula to find the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values of A, B, and C:
\[ x = \frac{a \pm \sqrt{(-a)^2 - 4 \cdot 1 \cdot (3 - a)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{a \pm \sqrt{a^2 - 12 + 4a}}{2} \]
\[ x = \frac{a \pm \sqrt{a^2 + 4a - 12}}{2} \]

### Step 4: Define the roots
Let:
- \( \alpha = \frac{a - \sqrt{a^2 + 4a - 12}}{2} \)
- \( \beta = \frac{a + \sqrt{a^2 + 4a - 12}}{2} \)

### Step 5: Analyze the condition that 1 lies between the roots
For 1 to lie between the roots, we need:
\[ \alpha < 1 < \beta \]

### Step 6: Set up inequalities
1. For \( \beta > 1 \):
\[ \frac{a + \sqrt{a^2 + 4a - 12}}{2} > 1 \]
This leads to:
\[ a + \sqrt{a^2 + 4a - 12} > 2 \]
\[ \sqrt{a^2 + 4a - 12} > 2 - a \]
Squaring both sides:
\[ a^2 + 4a - 12 > (2 - a)^2 \]
\[ a^2 + 4a - 12 > 4 - 4a + a^2 \]
Simplifying gives:
\[ 8a > 16 \]
\[ a > 2 \]

2. For \( \alpha < 1 \):
\[ \frac{a - \sqrt{a^2 + 4a - 12}}{2} < 1 \]
This leads to:
\[ a - \sqrt{a^2 + 4a - 12} < 2 \]
\[ -\sqrt{a^2 + 4a - 12} < 2 - a \]
Squaring both sides:
\[ a^2 + 4a - 12 < (2 - a)^2 \]
\[ a^2 + 4a - 12 < 4 - 4a + a^2 \]
Simplifying gives:
\[ 8a < 16 \]
\[ a < 2 \]

### Step 7: Combine the results
From the two inequalities derived:
1. \( a > 2 \)
2. \( a < 2 \)

This means that the only valid solution for \( a \) is:
\[ a \in (2, \infty) \]

### Final Answer
Thus, the values of \( a \) that satisfy the condition that 1 lies between the roots \( \alpha \) and \( \beta \) are:
\[ a \in (2, \infty) \]