Let's start by analyzing the given equation:
[tex]\[
q^2 + x = 0
\][/tex]
We are given that [tex]\( q = -i \sqrt{2} \)[/tex] is a solution to this equation. First, let's substitute [tex]\( q = -i \sqrt{2} \)[/tex] into the equation and solve for [tex]\( x \)[/tex]:
[tex]\[
(-i \sqrt{2})^2 + x = 0
\][/tex]
We need to calculate [tex]\( (-i \sqrt{2})^2 \)[/tex]:
[tex]\[
(-i \sqrt{2})^2 = (-i \sqrt{2}) \cdot (-i \sqrt{2}) = (-i) \cdot (-i) \cdot (\sqrt{2} \cdot \sqrt{2})
\][/tex]
Since [tex]\( (-i) \cdot (-i) = i^2 = -1 \)[/tex] and [tex]\( \sqrt{2} \cdot \sqrt{2} = 2 \)[/tex], we have:
[tex]\[
(-i \sqrt{2})^2 = (-1) \cdot 2 = -2
\][/tex]
So the equation becomes:
[tex]\[
-2 + x = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives us:
[tex]\[
x = 2
\][/tex]
Thus, the given equation is:
[tex]\[
q^2 - 2 = 0
\][/tex]
Next, let’s solve this quadratic equation for [tex]\( q \)[/tex]. We can rewrite it in standard form:
[tex]\[
q^2 = 2
\][/tex]
To find the solutions, take the square root of both sides:
[tex]\[
q = \pm \sqrt{2}
\][/tex]
Since we need to account for the imaginary unit [tex]\( i \)[/tex], the two solutions will be:
[tex]\[
q = \pm i \sqrt{2}
\][/tex]
So, we have two solutions:
[tex]\[
q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2}
\][/tex]
Therefore, the two solutions to this equation are:
[tex]\[
q = -i \sqrt{2} \text{ and } q = i \sqrt{2}
\][/tex]
Expressed in a simplified, rationalized form:
[tex]\[
q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2}
\][/tex]
