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The problem you uploaded asks for two things:

1. **Solve the equation:**
   \[
   x^2 - \frac{9}{x^2} = 10
   \]
   for \( x \) such that the inequality \( x^3 - 3x > 0 \) is also satisfied.

2. **Show that the function:**
   \[
   y = x^3 - 3x
   \]
   is increasing at \( x = 1 \).

### Step 1: Solving the equation

The equation is:
\[
x^2 - \frac{9}{x^2} = 10
\]
Multiplying both sides by \( x^2 \) to eliminate the fraction gives:
\[
x^4 - 9 = 10x^2
\]
Rearranging this, we get a quadratic equation in terms of \( x^2 \):
\[
x^4 - 10x^2 - 9 = 0
\]
Let \( z = x^2 \). This simplifies to:
\[
z^2 - 10z - 9 = 0
\]
We can solve this quadratic using the quadratic formula:
\[
z = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-9)}}{2(1)}
\]
\[
z = \frac{10 \pm \sqrt{100 + 36}}{2} = \frac{10 \pm \sqrt{136}}{2}
\]
\[
z = \frac{10 \pm 2\sqrt{34}}{2}
\]
\[
z = 5 \pm \sqrt{34}
\]
Thus, \( x^2 = 5 + \sqrt{34} \) or \( x^2 = 5 - \sqrt{34} \).

Since \( \sqrt{34} \) is approximately 5.83, the second solution \( 5 - \sqrt{34} \) is negative, which is not possible for \( x^2 \). Hence, the only valid solution is:
\[
x^2 = 5 + \sqrt{34}
\]
Taking the square root of both sides:
\[
x = \pm \sqrt{5 + \sqrt{34}}
\]

### Step 2: Checking the inequality \( x^3 - 3x > 0 \)

The inequality to check is \( x^3 - 3x > 0 \). Factorizing this, we get:
\[
x(x^2 - 3) > 0
\]
The critical points occur when \( x = 0 \) or \( x = \pm \sqrt{3} \).

Testing intervals:
- For \( x \in (-\infty, -\sqrt{3}) \), the expression is positive.
- For \( x \in (-\sqrt{3}, 0) \), the expression is negative.
- For \( x \in (0, \sqrt{3}) \), the expression is negative.
- For \( x \in (\sqrt{3}, \infty) \), the expression is positive.

Thus, \( x^3 - 3x > 0 \) when \( x > \sqrt{3} \) or \( x < -\sqrt{3} \).

### Step 3: Verifying at \( x = 1 \) that the function is increasing

To show that \( y = x^3 - 3x \) is increasing at \( x = 1 \), we take the derivative:
\[
y' = \frac{d}{dx}(x^3 - 3x) = 3x^2 - 3
\]
At \( x = 1 \):
\[
y'(1) = 3(1)^2 - 3 = 3 - 3 = 0
\]
This means the function is not increasing or decreasing at \( x = 1 \), as the derivative equals zero (it has a critical point).

Would you like more details or have any further questions?

Here are 5 related questions to explore further:
1. How do we solve quadratic equations in general?
2. What does the sign of the derivative tell us about a function's behavior?
3. What is the graphical interpretation of the function \( y = x^3 - 3x \)?
4. How can we determine intervals where a function is increasing or decreasing?
5. How do we solve equations involving higher powers (quartic equations)?

**Tip:** Always verify the domain of solutions when dealing with square roots or fractions in equations!

Consider the equation x^2 - 9/x^2 = 10. Solve for x such that the inequality x^3 - 3x > 0 is also satisfied. Additionally, show that y = x^3 - 3x is increasing at x = 1.

Learn how to solve the equation x^2 - 9/x^2 = 10 and verify the inequality x^3 - 3x > 0. Additionally, explore how to show that the function y = x^3 - 3x is increasing at x = 1 using derivatives. Suitable for high school students.

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