Context: For real x, the number of solutions to the equation (2√2 + √7)x + (2√2 – √7)x = 30 is:
Answer question

We are tasked with finding the number of solutions to the equation:

\[
(2\sqrt{2} + \sqrt{7})^x + (2\sqrt{2} - \sqrt{7})^x = 30
\]

### Step-by-Step Solution:

1. Let \( a = 2\sqrt{2} + \sqrt{7} \) and \( b = 2\sqrt{2} - \sqrt{7} \).
   Therefore, the given equation becomes:
   \[
   a^x + b^x = 30
   \]
   
2. First, let's examine the properties of \( a \) and \( b \). Note that \( a \) and \( b \) are conjugates, meaning:
   \[
   a \cdot b = (2\sqrt{2} + \sqrt{7})(2\sqrt{2} - \sqrt{7}) = (2\sqrt{2})^2 - (\sqrt{7})^2 = 8 - 7 = 1
   \]
   Hence, \( a \cdot b = 1 \).

3. This means that \( b = \frac{1}{a} \), so the equation becomes:
   \[
   a^x + \frac{1}{a^x} = 30
   \]
   Let \( y = a^x \). Thus, the equation transforms into:
   \[
   y + \frac{1}{y} = 30
   \]
   
4. Multiply both sides of the equation by \( y \):
   \[
   y^2 + 1 = 30y
   \]
   Rearranging terms:
   \[
   y^2 - 30y + 1 = 0
   \]
   
5. Now, solve this quadratic equation using the quadratic formula:
   \[
   y = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(1)(1)}}{2(1)}
   \]
   \[
   y = \frac{30 \pm \sqrt{900 - 4}}{2}
   \]
   \[
   y = \frac{30 \pm \sqrt{896}}{2}
   \]
   \[
   y = \frac{30 \pm 29.933}{2}
   \]

6. This gives two possible values for \( y \):
   \[
   y_1 = \frac{30 + 29.933}{2} = 29.9665, \quad y_2 = \frac{30 - 29.933}{2} = 0.0335
   \]

7. Since \( y = a^x \), we need \( y > 0 \), which is true for both solutions. Now, solve for \( x \) using each value of \( y \):
   \[
   a^x = 29.9665 \quad \text{or} \quad a^x = 0.0335
   \]

8. Both cases are valid and yield distinct values of \( x \), meaning there are two solutions.

### Conclusion:
The equation has **2 real solutions**.

Would you like further clarification or details on any of these steps?

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Here are some related questions:
1. What is the significance of conjugate pairs in solving such equations?
2. How does the quadratic formula help solve exponential equations?
3. Can you generalize this method for different types of equations?
4. What are the conditions for a quadratic equation to have real solutions?
5. How would the solution change if the right-hand side of the equation was different?

**Tip:** Always check the properties of the terms (like conjugates or factorizations) when dealing with complex-looking expressions, as they often simplify the problem.

For real x, the number of solutions to the equation (2√2 + √7)^x + (2√2 – √7)^x = 30 is:

Discover how to solve the equation (2√2 + √7)^x + (2√2 – √7)^x = 30, utilizing key concepts such as conjugates, quadratic equations, and the quadratic formula. This problem is suitable for students in Grades 10-12, focusing on advanced algebra and exponential equations.

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