The equation x^2–px–60 = 0, has two roots ‘a’ and ‘b’ such that (a –b) = 17 and p > 0. If a series starts with ‘p’ such that the consecutive terms are 4 more than the preceding term is formed, then find the product of 2nd and 4th terms of such series.

To solve this problem, we need to break it down into two main parts:

1. **Finding the value of \( p \):**

   The given quadratic equation is \( x^2 - px - 60 = 0 \), with roots \( a \) and \( b \). 

   By Vieta's formulas, we know:
   \[
   a + b = p
   \]
   \[
   ab = -60
   \]
   Also, it's given that \( |a - b| = 17 \). Since \( p > 0 \), we consider the cases where \( a > b \) or \( b > a \).

   From the equation \( |a - b| = 17 \), let's assume \( a > b \), so:
   \[
   a - b = 17
   \]

   We can solve for \( a \) and \( b \) using the equations:
   \[
   a + b = p
   \]
   \[
   a - b = 17
   \]

   Adding these two equations:
   \[
   2a = p + 17 \implies a = \frac{p + 17}{2}
   \]

   Subtracting the second from the first:
   \[
   2b = p - 17 \implies b = \frac{p - 17}{2}
   \]

   Now substituting these expressions into the product equation \( ab = -60 \):

   \[
   \left(\frac{p + 17}{2}\right) \left(\frac{p - 17}{2}\right) = -60
   \]

   Simplifying the left side:
   \[
   \frac{(p + 17)(p - 17)}{4} = -60
   \]

   This is a difference of squares:
   \[
   \frac{p^2 - 289}{4} = -60
   \]

   Multiplying both sides by 4:
   \[
   p^2 - 289 = -240
   \]

   Adding 289 to both sides:
   \[
   p^2 = 49
   \]

   Taking the square root of both sides:
   \[
   p = 7 \quad (\text{since } p > 0)
   \]

2. **Finding the product of the 2nd and 4th terms of the series:**

   The series starts with \( p = 7 \) and each consecutive term is 4 more than the preceding term. Thus, the series is:

   \[
   7, 11, 15, 19, \ldots
   \]

   The 2nd term is \( 11 \) and the 4th term is \( 19 \).

   The product of the 2nd and 4th terms is:
   \[
   11 \times 19 = 209
   \]

So, the product of the 2nd and 4th terms of the series is **209**.

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions:
1. How can you use Vieta's formulas to solve other quadratic equations?
2. What is the significance of the difference of squares in solving equations?
3. How do you find the roots of a quadratic equation if the discriminant is zero?
4. What are arithmetic sequences, and how are they related to this problem?
5. How would the solution change if \( p \) were negative?

**Tip:** Always check if a quadratic equation can be factored easily to simplify the solution process.

Learn how to solve the quadratic equation x^2 - px - 60 using Vieta's formulas. This problem involves finding the roots 'a' and 'b' where |a - b| = 17 and p > 0, and then calculating the product of the 2nd and 4th terms of an arithmetic series starting with p. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation