Final answer:
The solutions to the equation x^4 - 9x^2 + 8 = 0 are found using u substitution, with u = x^2. After factoring the quadratic in terms of u, we find the solutions for x are ±√8 and ±1.
Step-by-step explanation:
To solve the equation x^4 - 9x^2 + 8 = 0 using u substitution, we first recognize this as a quadratic equation in disguise. To make it more apparent, let's use substitution: let u = x^2. Substituting u into the equation gives us:
u^2 - 9u + 8 = 0. This is now a standard quadratic equation that we can factor: (u - 8)(u - 1) = 0.
Setting each factor equal to zero gives us the solutions for u: u = 8 and u = 1. We need to revert back to our original variable x, so we substitute back to get x^2 = 8 and x^2 = 1. This gives us four possible values for x: x = ±√8 and x = ±1.
Therefore, the solutions for the original equation are x = √8, x = -√8, x = 1, and x = -1.