To fill in the missing number so that [tex]\( q = -9i \)[/tex] is a solution of the equation, we need to calculate [tex]\( q^2 \)[/tex].
Given:
[tex]\[ q = -9i \][/tex]
First, we calculate [tex]\( q^2 \)[/tex]:
[tex]\[ q^2 = (-9i)^2 \][/tex]
We know that [tex]\( i \)[/tex] is the imaginary unit where [tex]\( i^2 = -1 \)[/tex]. Therefore:
[tex]\[ (-9i)^2 = (-9)^2 \times (i^2) \][/tex]
[tex]\[ (-9i)^2 = 81 \times (-1) \][/tex]
[tex]\[ (-9i)^2 = -81 \][/tex]
So, the equation [tex]\( q^2 \)[/tex] becomes:
[tex]\[ q^2 = -81 \][/tex]
Now, we need to find the two solutions to the equation:
[tex]\[ q^2 = -81 \][/tex]
We already know that one solution is [tex]\( q = -9i \)[/tex].
To find the second solution, we consider the properties of square roots in the complex number domain. Since [tex]\( (-9i)^2 = -81 \)[/tex], the other solution to [tex]\( q^2 = -81 \)[/tex] would be the positive counterpart of the imaginary number:
[tex]\[ q = 9i \][/tex]
Thus, the two solutions to the equation [tex]\( q^2 = -81 \)[/tex] are:
[tex]\[ q = -9i \][/tex]
[tex]\[ q = 9i \][/tex]
In summary:
Fill in the missing number so that [tex]\( q = -9i \)[/tex] is a solution of the equation:
[tex]\[ q^2 = -81 \][/tex]
What are the two solutions to this equation?
[tex]\[ q = -9i \text{ and } q = 9i \][/tex]
