If ` alpha and beta` are the roots of the equation ` x^(2)-px +16=0` , such that ` alpha^(2)+beta^(2)=9`, then the value of p is

Options

A. `+-sqrt6`
B. `+-sqrt41`
C. `+-8`
D. `+-7`

Answer

To find the value of \( p \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - px + 16 = 0 \) and \( \alpha^2 + \beta^2 = 9 \), we can follow these steps:

### Step 1: Identify the relationships from the roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = p \)
- The product of the roots \( \alpha \beta = 16 \)

### Step 2: Express \( \alpha^2 + \beta^2 \) in terms of \( p \) and \( \alpha \beta \)
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we have:
\[ \alpha^2 + \beta^2 = p^2 - 2 \cdot 16 \]
This simplifies to:
\[ \alpha^2 + \beta^2 = p^2 - 32 \]

### Step 3: Set up the equation using the given condition
We know from the problem that \( \alpha^2 + \beta^2 = 9 \). Therefore, we can set up the equation:
\[ p^2 - 32 = 9 \]

### Step 4: Solve for \( p^2 \)
Rearranging the equation gives:
\[ p^2 = 9 + 32 \]
\[ p^2 = 41 \]

### Step 5: Find the value of \( p \)
Taking the square root of both sides, we get:
\[ p = \sqrt{41} \quad \text{or} \quad p = -\sqrt{41} \]

### Final Answer
Thus, the values of \( p \) are \( \sqrt{41} \) and \( -\sqrt{41} \).

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