To find the solutions of the equation [tex]\( x^4 + 3x^2 + 2 = 0 \)[/tex], we can use a substitution method. Let's go through this step by step.
1. Substitution:
Let [tex]\( u = x^2 \)[/tex]. This transforms the equation into a quadratic form:
[tex]\[
u^2 + 3u + 2 = 0
\][/tex]
2. Solve the quadratic equation:
This is a standard quadratic equation which can be solved using either factoring, completing the square, or the quadratic formula. We will factor it:
[tex]\[
u^2 + 3u + 2 = 0
\][/tex]
After factoring, we get:
[tex]\[
(u + 1)(u + 2) = 0
\][/tex]
Setting each factor to zero gives us the solutions for [tex]\( u \)[/tex]:
[tex]\[
u + 1 = 0 \quad \Rightarrow \quad u = -1
\][/tex]
[tex]\[
u + 2 = 0 \quad \Rightarrow \quad u = -2
\][/tex]
3. Back-substitute [tex]\( u = x^2 \)[/tex]:
These solutions for [tex]\( u \)[/tex] translate back to [tex]\( x \)[/tex] as follows:
- For [tex]\( u = -1 \)[/tex]:
[tex]\[
x^2 = -1 \quad \Rightarrow \quad x = \pm i
\][/tex]
- For [tex]\( u = -2 \)[/tex]:
[tex]\[
x^2 = -2 \quad \Rightarrow \quad x = \pm i\sqrt{2}
\][/tex]
4. Compile all solutions:
The solutions we found for [tex]\( x \)[/tex] are:
[tex]\[
x = \pm i, \quad x = \pm i \sqrt{2}
\][/tex]
Therefore, the complete set of solutions for the equation [tex]\( x^4 + 3x^2 + 2 = 0 \)[/tex] is:
[tex]\[
x = \pm i, \quad x = \pm i \sqrt{2}
\][/tex]
Given the solution options:
- [tex]\( x = \pm i \sqrt{2} \)[/tex] and [tex]\( x = \pm 1 \)[/tex]
- [tex]\( x = \pm i \sqrt{2} \)[/tex] and [tex]\( x = \pm i \)[/tex]
- [tex]\( x = \pm \sqrt{2} \)[/tex] and [tex]\( x = \pm i \)[/tex]
- [tex]\( x = \pm \sqrt{2} \)[/tex] and [tex]\( x = \pm 1 \)[/tex]
The correct set of solutions is:
[tex]\[
x = \pm i \sqrt{2} \quad \text{and} \quad x = \pm i
\][/tex]
So, the correct answer is:
[tex]\[
2
\][/tex]
