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

by

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:

    2. 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:

    3. (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!

Leave a Comment