We are given the quadratic equation:

$x^2 - (p - 2)x - 6 = 0$

The roots of this equation are $m$ and $n$, and we are also given that $m^2 + 2mn + n^2 = 9$. Let's solve for $p$.

Step 1: Use Vieta's Formulas

From Vieta's formulas, for a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are given by: $m + n = -\frac{b}{a}, \quad mn = \frac{c}{a}.$ For our equation $x^2 - (p - 2)x - 6 = 0$, we have:

• Sum of roots: $m + n = p - 2$,
• Product of roots: $mn = -6$.

Step 2: Use the given condition

We are also given the condition: $m^2 + 2mn + n^2 = 9.$ We can rewrite this as: $(m + n)^2 = m^2 + 2mn + n^2 = 9.$

Step 3: Substitute the sum and product of the roots

We already know:

• $m + n = p - 2$,
• $mn = -6$.

Substitute $m + n = p - 2$ into $(m + n)^2 = 9$: $(p - 2)^2 = 9.$

Step 4: Solve for $p$

Expand the equation: $(p - 2)^2 = 9 \quad \Rightarrow \quad p - 2 = \pm 3.$ This gives two possible solutions:

1. $p - 2 = 3 \quad \Rightarrow \quad p = 5$,
2. $p - 2 = -3 \quad \Rightarrow \quad p = -1$.

Step 5: Verify the solutions

We need to check both solutions with the condition $mn = -6$.

For $p = 5$:

• $m + n = 5 - 2 = 3$.
• Solve the quadratic equation $t^2 - 3t - 6 = 0$, and the discriminant is $\Delta = 9 + 24 = 33$, which yields real roots, so $p = 5$ is valid.

For $p = -1$:

• $m + n = -1 - 2 = -3$.
• Solve the quadratic equation $t^2 + 3t - 6 = 0$, and the discriminant is also $33$, which also yields real roots, so $p = -1$ is also valid.

Conclusion:

The possible values of $p$ are $5$ and $-1$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does Vieta's formulas apply to general quadratic equations?
2. How do you solve quadratic equations using the discriminant?
3. What are the possible cases when $m^2 + 2mn + n^2 = 9$?
4. How does the discriminant help determine the nature of roots?
5. What other techniques can be used to verify the roots of a quadratic equation?

Tip: Always check your solutions by substituting them back into the original equation or conditions to ensure they satisfy all given constraints.