Solving the Equation (x^2 + 1)^2  6(x^2 + 1) + 9 = 0
This equation might look intimidating at first glance, but we can solve it using a simple substitution.
Substitution Method

Let's make a substitution: Let y = x² + 1. This will simplify the equation.

Substitute: Now, our equation becomes: y²  6y + 9 = 0

Solve for y: This is a quadratic equation that can be easily solved by factoring: (y  3)(y  3) = 0 Therefore, y = 3

Substitute back: Now, replace y with x² + 1: x² + 1 = 3

Solve for x:
x² = 2 x = ±√2
The Solutions
Therefore, the solutions to the equation (x² + 1)²  6(x² + 1) + 9 = 0 are:
x = √2 and x = √2
Conclusion
By using a simple substitution, we were able to transform a seemingly complex equation into a familiar quadratic equation. This demonstrates the power of algebraic manipulation in simplifying mathematical problems.