To solve for the missing number in the equation where [tex]\( q = -i \sqrt{2} \)[/tex]:
[tex]\[
q^2 + \square = 0
\][/tex]
let's first square [tex]\( q \)[/tex]:
[tex]\[
q = -i \sqrt{2}
\][/tex]
Squaring both sides, we get:
[tex]\[
q^2 = (-i \sqrt{2})^2
\][/tex]
We know that:
[tex]\[
(-i \sqrt{2})^2 = (i^2)(\sqrt{2})^2 = (-1)(2) = -2
\][/tex]
So:
[tex]\[
q^2 = -2
\][/tex]
To make the equation [tex]\( q^2 + \square = 0 \)[/tex] true, we need:
[tex]\[
q^2 + 2 = 0
\][/tex]
Thus, the missing number is [tex]\( 2 \)[/tex]. To verify, substitute:
[tex]\[
q^2 + 2 = 0 \implies -2 + 2 = 0
\][/tex]
Next, we need to find the two solutions of the equation:
[tex]\[
q^2 + 2 = 0
\][/tex]
First, rearrange it to standard form:
[tex]\[
q^2 = -2
\][/tex]
Taking the square root of both sides:
[tex]\[
q = \pm \sqrt{-2}
\][/tex]
Recall that the square root of a negative number involves the imaginary unit [tex]\( i \)[/tex]:
[tex]\[
\sqrt{-2} = \sqrt{2} \cdot i
\][/tex]
Thus, the two solutions are:
[tex]\[
q = i \sqrt{2} \quad \text{and} \quad q = -i \sqrt{2}
\][/tex]
In summary, the two solutions in simplified, rationalized form are:
[tex]\[
q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2}
\][/tex]
