𝑥𝑥1 𝑥𝑥1
b. If x is a real number satisfying x3 +1/x3 = 2√5, determine the exact value of x2+1/x2.

To determine the exact value of \( x^2 + \frac{1}{x^2} \) given that \( x \) is a real number satisfying \( x^3 + \frac{1}{x^3} = 2\sqrt{5} \), let's follow these steps:

1. **Let \( y = x + \frac{1}{x} \)**: We will first express \( x^3 + \frac{1}{x^3} \) in terms of \( y \).

2. **Cube \( y \)**:
   \[
   y^3 = \left( x + \frac{1}{x} \right)^3
   \]
   Expanding the right-hand side, we get:
   \[
   y^3 = x^3 + 3x \cdot \frac{1}{x} \cdot \left( x + \frac{1}{x} \right) + \frac{1}{x^3}
   \]
   \[
   y^3 = x^3 + 3\left( x + \frac{1}{x} \right) + \frac{1}{x^3}
   \]
   \[
   y^3 = x^3 + \frac{1}{x^3} + 3y
   \]

3. **Given equation \( x^3 + \frac{1}{x^3} = 2\sqrt{5} \)**:
   Substitute this into the previous result:
   \[
   y^3 = 2\sqrt{5} + 3y
   \]
   This simplifies to:
   \[
   y^3 - 3y - 2\sqrt{5} = 0
   \]

4. **Solve for \( y \)**:
   This is a cubic equation in \( y \). We need to find a value of \( y \) that satisfies the equation. Checking possible rational roots:
   \[
   y = \sqrt{5} \text{ is a possible root}
   \]
   Substitute \( y = \sqrt{5} \) into the equation:
   \[
   (\sqrt{5})^3 - 3\sqrt{5} - 2\sqrt{5} = 0
   \]
   \[
   5\sqrt{5} - 5\sqrt{5} = 0
   \]
   It holds true, so \( y = \sqrt{5} \).

5. **Find \( x^2 + \frac{1}{x^2} \)**:
   From the identity \( y^2 = x^2 + 2 + \frac{1}{x^2} \):
   \[
   (\sqrt{5})^2 = x^2 + \frac{1}{x^2} + 2
   \]
   \[
   5 = x^2 + \frac{1}{x^2} + 2
   \]
   \[
   x^2 + \frac{1}{x^2} = 3
   \]

Therefore, the exact value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{3} \).

Would you like more details or have any questions about this solution?

Here are some more questions you might consider:
1. How do you solve cubic equations generally?
2. Can you explain the process of expanding \( (x + \frac{1}{x})^3 \)?
3. What are some other identities involving \( x + \frac{1}{x} \)?
4. How can you solve for real roots in polynomial equations?
5. What is the significance of rational root theorem in solving polynomial equations?

**Tip:** Always check for possible rational roots when dealing with polynomial equations, as they can simplify the solving process significantly.

Discover the exact value of x^2 + 1/x^2 when x^3 + 1/x^3 equals 2√5. This problem involves solving a cubic equation and applying identities related to x and 1/x. 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