How can I solve the equation x^4 + 9x^2 + 8 = 0 using u substitution?

To solve the equation x⁴ + 9x² + 8 = 0 using u substitution, we can follow these steps:

1. Substitution: Let u = x². This means that x⁴ can be expressed as u². Therefore, we rewrite the original equation in terms of u:

u² + 9u + 8 = 0

2. Factoring the Quadratic: Next, we need to factor the quadratic equation. We can look for two numbers that multiply to 8 and add up to 9. The numbers are 1 and 8. Thus, we can factor it as:

(u + 1)(u + 8) = 0

3. Finding the Roots: Now, we set each factor equal to zero and solve for u:
• u + 1 = 0:
• u = -1
• u + 8 = 0:
• u = -8
4. Back Substitution: Since we defined u = x², we substitute back to find the values of x:
• x² = -1: This does not yield real solutions, but we can express the solutions in terms of imaginary numbers:
• Thus, x = ±i
• x² = -8: Similarly, this does not yield real solutions, so:
• Therefore, x = ±2i√2
5. Final Solutions: The final solutions to the equation x⁴ + 9x² + 8 = 0 are:
• x = ±i
• x = ±2i√2

In conclusion, by using u substitution, we transformed the original equation into a simpler quadratic form, facilitated the solving process, and eventually determined the complex solutions for x. This illustrates a powerful method in algebraic problem-solving!