To solve the equation x4 + 9x2 + 8 = 0, we can use a substitution method. Let’s set u = x2. This transforms our original equation into a quadratic format.
Substituting u gives us:
u2 + 9u + 8 = 0
Now, we can factor this quadratic equation. We are looking for two numbers that multiply to 8 (the constant term) and add to 9 (the coefficient of the linear term). The numbers 1 and 8 work:
(u + 1)(u + 8) = 0
Setting each factor to zero gives us:
- u + 1 = 0 → u = -1
- u + 8 = 0 → u = -8
Now we’ll substitute back x2 for u:
x2 = -1 and x2 = -8
Since both equations have negative results, we will have complex solutions:
- For x2 = -1: x = ±i
- For x2 = -8: x = ±2i√2
Thus, the final solutions are:
- x = i
- x = -i
- x = 2i√2
- x = -2i√2
To summarize, using the substitution method has allowed us to find the complex solutions to the original equation.