If `alpha, beta` are the roots of `x^(2) + 7x + 3 = 0` then `(alpha – 1)^(2) + (beta – 1)^(2)`=

Options

A. 59
B. 69
C. 39
D. 49

Answer

To solve the problem, we need to find the value of \((\alpha - 1)^2 + (\beta - 1)^2\), where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + 7x + 3 = 0\).

### Step 1: Find the roots of the quadratic equation
We can use the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \(x^2 + 7x + 3 = 0\), we have:
- \(a = 1\)
- \(b = 7\)
- \(c = 3\)

Substituting these values into the formula:
\[ \alpha, \beta = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 7^2 - 4 \cdot 1 \cdot 3 = 49 - 12 = 37 \]
Now substituting back:
\[ \alpha, \beta = \frac{-7 \pm \sqrt{37}}{2} \]

### Step 2: Calculate \((\alpha - 1)^2 + (\beta - 1)^2\)
Using the identity:
\[ (\alpha - 1)^2 + (\beta - 1)^2 = \alpha^2 - 2\alpha + 1 + \beta^2 - 2\beta + 1 \]
This simplifies to:
\[ = \alpha^2 + \beta^2 - 2(\alpha + \beta) + 2 \]

### Step 3: Find \(\alpha + \beta\) and \(\alpha^2 + \beta^2\)
From Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} = -7 \]
\[ \alpha \beta = \frac{c}{a} = 3 \]
Now, we can find \(\alpha^2 + \beta^2\) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values:
\[ \alpha^2 + \beta^2 = (-7)^2 - 2 \cdot 3 = 49 - 6 = 43 \]

### Step 4: Substitute back into the equation
Now, substituting \(\alpha + \beta\) and \(\alpha^2 + \beta^2\) back into our expression:
\[ (\alpha - 1)^2 + (\beta - 1)^2 = 43 - 2(-7) + 2 \]
Calculating:
\[ = 43 + 14 + 2 = 59 \]

### Final Answer
Thus, the value of \((\alpha - 1)^2 + (\beta - 1)^2\) is:
\[ \boxed{59} \]