x²-(p-2)x-6=0 has roots of my and n, where m²+2mn+n²=9, find the value of p

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.

x²-(p-2)x-6=0 has roots of m and n, where m²+2mn+n²=9, find the value of p

Learn how to solve the quadratic equation x²-(p-2)x-6=0 where the roots are m and n, and m²+2mn+n²=9. This problem uses Vieta's formulas and the discriminant to find the values of p. Suitable for students in Grades 9-11.

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