What are the solutions of the equation x^4 + 3x^2 + 2 = 0? Use u substitution to solve.

To solve the equation x⁴ + 3x² + 2 = 0 using u substitution, we can start by letting u = x². This means that x⁴ = u². By substituting these into the original equation, we rewrite it as:

u² + 3u + 2 = 0

Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to 2 (the constant term) and add up to 3 (the coefficient of u). The numbers that satisfy these conditions are 1 and 2. Thus, we can factor the equation as:

(u + 1)(u + 2) = 0

Setting each factor equal to zero gives us:

• u + 1 = 0 → u = -1
• u + 2 = 0 → u = -2

Now, recall that we substituted u = x². Therefore, we will replace u back with x²:

• x² = -1
• x² = -2

Both of these equations lead us to complex solutions since we cannot have a square root of a negative number in the realm of real numbers.

For x² = -1, the solutions are:

• x = i (where i is the imaginary unit)
• x = -i

For x² = -2, the solutions are:

• x = 0101 i extsqrt{2}
• x = 0101 – i extsqrt{2}

In conclusion, the solutions to the equation x⁴ + 3x² + 2 = 0 are:

• x = i
• x = -i
• x = extsqrt{2} i
• x = – extsqrt{2} i