To find the value to fill in the missing number so that [tex]\( r = -5i \)[/tex] is a solution of the equation [tex]\( r^2 = \square \)[/tex], and determine the two solutions to this equation, we proceed as follows:
1. Substitute [tex]\( r = -5i \)[/tex] into the equation [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = (-5i)^2 \][/tex]
2. Compute [tex]\( (-5i)^2 \)[/tex]:
- First, square the coefficient: [tex]\( (-5)^2 = 25 \)[/tex]
- Then recall that [tex]\( i^2 = -1 \)[/tex], since [tex]\( i \)[/tex] is the imaginary unit.
- Now multiply these results: [tex]\( 25 \times (-1) = -25 \)[/tex]
Therefore,
[tex]\[ r^2 = -25 \][/tex]
3. The equation thus becomes:
[tex]\[ r^2 = -25 \][/tex]
4. To find the solutions to [tex]\( r^2 = -25 \)[/tex], we solve for [tex]\( r \)[/tex] by taking the square root of both sides:
[tex]\[ r = \pm \sqrt{-25} \][/tex]
5. Simplify the square root of the negative number:
- Recall the property of square roots of negative numbers: [tex]\( \sqrt{-a} = i\sqrt{a} \)[/tex]
- Here, [tex]\( \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \)[/tex]
Thus, the solutions are:
[tex]\[ r = 5i \quad \text{and} \quad r = -5i \][/tex]
### Summary:
- The missing number that completes the equation is [tex]\( -25 \)[/tex].
- The two solutions to the equation [tex]\( r^2 = -25 \)[/tex] are:
[tex]\[ r = -5i \quad \text{and} \quad r = 5i \][/tex]
This provides the complete information in simplified, rationalized form.
