If the roots `alpha,beta` of the equation `x^(2)-px+16=0` satisfy the relation `alpha^(2)+beta^(2)=4`, then the value(s) of p is/are

Options

A. 6 only
B. `-6` only
C. 6 or `-6`
D. 8 or `-6`

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Identify the coefficients from the quadratic equation
The given quadratic equation is:
\[ x^2 - px + 16 = 0 \]

From this equation, we can identify:
- \( a = 1 \)
- \( b = -p \)
- \( c = 16 \)

### Step 2: Use Vieta's formulas to find the sum and product of the roots
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-p}{1} = p \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{16}{1} = 16 \)

### Step 3: Use the relation involving the roots
We are given that:
\[ \alpha^2 + \beta^2 = 4 \]

### Step 4: 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 we found:
\[ \alpha^2 + \beta^2 = p^2 - 2 \cdot 16 \]
\[ \alpha^2 + \beta^2 = p^2 - 32 \]

### Step 5: Set up the equation
Now, we can set this equal to the given value:
\[ p^2 - 32 = 4 \]

### Step 6: Solve for \( p^2 \)
Rearranging the equation gives:
\[ p^2 = 4 + 32 \]
\[ p^2 = 36 \]

### Step 7: Find the values of \( p \)
Taking the square root of both sides:
\[ p = \pm 6 \]

### Final Answer
The values of \( p \) are:
\[ p = 6 \quad \text{or} \quad p = -6 \]

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