https://file.aiquickdraw.com/mathbot/file/1725580395472h8rdbspx.jpg

It seems that only part of the text was extracted. I will now focus on the equation visible in the image to solve it manually.

The equation appears to be:
\[
\frac{1}{2x^2} - 9x^4 + 5 = 0
\]

Let's solve this equation step by step.

### 1. Move all terms to one side:
\[
\frac{1}{2x^2} = 9x^4 - 5
\]

Multiply both sides by \( 2x^2 \) to eliminate the fraction:
\[
1 = 2x^2(9x^4 - 5)
\]

Now, simplify:
\[
1 = 18x^6 - 10x^2
\]

### 2. Rearrange the equation:
\[
18x^6 - 10x^2 - 1 = 0
\]

This is a quartic equation. Let \( y = x^2 \), transforming the equation into a cubic in \( y \):
\[
18y^3 - 10y - 1 = 0
\]

Now, I will solve this cubic equation for \( y \).The solutions to the cubic equation \(18y^3 - 10y - 1 = 0\) are complex and not simple real numbers. This suggests that the original equation might not have real solutions. However, to further verify this, we could analyze the discriminant of the equation to confirm the absence of real roots.

Would you like to explore this further, or try solving it numerically? Let me know if you want additional details or clarifications!

Here are five related questions to deepen your understanding:
1. How can you determine whether a polynomial equation has real or complex solutions?
2. What methods are available for solving cubic equations, both analytically and numerically?
3. How does the substitution \( y = x^2 \) help simplify higher-degree polynomial equations?
4. What are the conditions for an equation to have real, multiple, or complex roots?
5. How can graphing help in visually determining the nature of solutions to a polynomial?

**Tip**: When faced with complex solutions, graphing the function can offer a visual perspective on where the real parts lie (if any).

Find the real solutions of the following equation: \(\frac{1}{2x^2} - 9x^4 + 5 = 0\)

Discover how to solve the rational equation \(\frac{1}{2x^2} - 9x^4 + 5 = 0\) by transforming it into a cubic equation. This problem covers advanced algebraic concepts including discriminant analysis for cubic equations and is suitable for students in Grades 11-12.

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