To solve the equation x^4 + 3x^2 + 2 = 0 using u substitution, we first transform the equation to make it simpler to work with.
We start by letting u = x^2. Consequently, the equation can be rewritten in terms of u:
u^2 + 3u + 2 = 0
This is now a standard quadratic equation. We can solve it using the factoring method.
To factor the quadratic, we look for two numbers that multiply to 2 (the constant term) and add up to 3 (the coefficient of the linear term). The numbers that meet these criteria are 1 and 2.
Thus, we can factor the quadratic as:
(u + 1)(u + 2) = 0
Next, we set each factor equal to zero to find the values of u:
u + 1 = 0 → u = -1
u + 2 = 0 → u = -2
Now, we substitute back x^2 for u:
x^2 = -1 → x = ±i (where i is the imaginary unit)
x^2 = -2 → x = ±√2i
The solutions to the original equation x^4 + 3x^2 + 2 = 0 are therefore:
x = i, -i, √2i, -√2i
In summary, we used u substitution to simplify the polynomial, solved for u, and then found the corresponding values for x.