If the roots of the equation `x^(2)+px+7=0` are denoted by `alpha and beta`, and `alpha^(2)+beta^(2)=22`, find the possible values of p.

Answer

To solve the problem, we start with the quadratic equation given:

\[ x^2 + px + 7 = 0 \]

The roots of this equation are denoted by \( \alpha \) and \( \beta \). We know two important relationships regarding the roots of a quadratic equation:

1. The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -p \)
2. The product of the roots \( \alpha \beta = \frac{c}{a} = 7 \)

We are also given that:

\[ \alpha^2 + \beta^2 = 22 \]

Now, we can use the identity for the sum of squares of the roots:

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

Substituting the expressions for the sum and product of the roots, we have:

\[ \alpha^2 + \beta^2 = (-p)^2 - 2(7) \]

This simplifies to:

\[ \alpha^2 + \beta^2 = p^2 - 14 \]

Since we know that \( \alpha^2 + \beta^2 = 22 \), we can set the two expressions equal to each other:

\[ p^2 - 14 = 22 \]

Now, we will solve for \( p^2 \):

1. Add 14 to both sides:

\[ p^2 = 22 + 14 \]
\[ p^2 = 36 \]

2. Now, take the square root of both sides:

\[ p = \pm \sqrt{36} \]
\[ p = \pm 6 \]

Thus, the possible values of \( p \) are:

\[ p = 6 \quad \text{or} \quad p = -6 \]

### Summary of Steps:
1. Identify the relationships for the sum and product of the roots.
2. Use the identity for the sum of squares of the roots.
3. Substitute the known values into the identity.
4. Solve for \( p^2 \).
5. Take the square root to find the possible values of \( p \).