https://file.aiquickdraw.com/mathbot/file/17263568319932v2zaeyh.jpg

The problem asks to solve for \( x_1 + k + x_2 \), where \( x_1 \) and \( x_2 \) are the roots of the quadratic equation:

\[
x^2 - (3k + 5)x + 2k + 3 = 0
\]

Additionally, \( x_1, k, x_2 \) form the first, second, and third terms of a geometric progression with a common ratio \( r \neq 1 \) and \( r \neq -1 \).

Let's break down the solution:

### 1. Use Vieta's Formulas:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us:
- The sum of the roots is \( x_1 + x_2 = -\frac{b}{a} \)
- The product of the roots is \( x_1 \cdot x_2 = \frac{c}{a} \)

In this case:
- \( a = 1 \), 
- \( b = -(3k + 5) \), 
- \( c = 2k + 3 \)

Thus, from Vieta's formulas:
\[
x_1 + x_2 = 3k + 5
\]
\[
x_1 \cdot x_2 = 2k + 3
\]

### 2. Geometric Progression:
Since \( x_1, k, x_2 \) form a geometric progression, we know that the square of the middle term equals the product of the other two terms:
\[
k^2 = x_1 \cdot x_2
\]

Substitute the expression for \( x_1 \cdot x_2 \):
\[
k^2 = 2k + 3
\]
This simplifies to the quadratic equation:
\[
k^2 - 2k - 3 = 0
\]
Factoring gives:
\[
(k - 3)(k + 1) = 0
\]
Thus, \( k = 3 \) or \( k = -1 \).

### 3. Check the common ratio:
We know that the common ratio \( r \neq 1 \) and \( r \neq -1 \). The ratio is given by:
\[
r = \frac{k}{x_1} = \frac{x_2}{k}
\]
Thus, we reject \( k = -1 \) because it would result in \( r = 1 \). Therefore, \( k = 3 \).

### 4. Solve for \( x_1 + k + x_2 \):
Substitute \( k = 3 \) into the expression for \( x_1 + x_2 \):
\[
x_1 + x_2 = 3(3) + 5 = 9 + 5 = 14
\]
Thus:
\[
x_1 + k + x_2 = 14 + 3 = 17
\]

### Final Answer:
The value of \( x_1 + k + x_2 \) is \( 17 \).

---

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do we determine the sum and product of the roots using Vieta’s formulas?
2. What are the properties of a geometric progression that we used here?
3. How would the solution change if \( r = 1 \)?
4. How do you factor a quadratic equation like \( k^2 - 2k - 3 = 0 \)?
5. What is the geometric significance of the condition \( r \neq 1 \) or \( r \neq -1 \)?

**Tip**: In quadratic equations, always look for connections between roots and coefficients through Vieta’s relations.

Jika x1, x2 adalah akar-akar persamaan kuadrat x^2 - (3k + 5)x + 2k + 3 = 0 dan x1, k, x2 merupakan suku pertama, kedua, dan ketiga suatu barisan geometri dengan rasio r ≠ 1 dan r ≠ -1, maka x1 + k + x2 = ?

Learn how to solve a quadratic equation where x1, k, and x2 form a geometric progression. This problem covers algebraic concepts, Vieta's formulas, and geometric progression properties. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation