The quadratic equation x^2 + px + q = 0 has roots -2 and 6. Find the value of p and of q

Given that the quadratic equation \(x^2 + px + q = 0\) has roots \(-2\) and \(6\), we can use Vieta's formulas, which relate the coefficients of the quadratic equation to the roots.

For a quadratic equation of the form \(x^2 + px + q = 0\), if \(\alpha\) and \(\beta\) are the roots, then:

1. The sum of the roots, \(\alpha + \beta = -p\).
2. The product of the roots, \(\alpha \times \beta = q\).

In this case, the roots are \(\alpha = -2\) and \(\beta = 6\).

1. **Sum of the roots:**
   \[
   \alpha + \beta = -2 + 6 = 4
   \]
   So, \(p = -4\).

2. **Product of the roots:**
   \[
   \alpha \times \beta = (-2) \times 6 = -12
   \]
   So, \(q = -12\).

Thus, the values of \(p\) and \(q\) are:
- \(p = -4\)
- \(q = -12\)

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How do Vieta's formulas work for higher-degree polynomials?
2. Can you derive the quadratic equation from the given roots \(-2\) and \(6\)?
3. What is the discriminant of this quadratic equation, and what does it indicate?
4. How can the quadratic formula be used to verify the roots?
5. What are the applications of quadratic equations in real-world problems?

**Tip:** Always double-check the signs when applying Vieta's formulas, as mistakes often occur there!

The quadratic equation x^2 + px + q = 0 has roots -2 and 6. Find the value of p and of q.

Learn how to solve the quadratic equation x^2 + px + q = 0 when given roots -2 and 6. This step-by-step solution uses Vieta's formulas to find the values of p and q. Perfect for students in Grades 8-10.

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