To solve for the missing number and determine the solutions to the equation [tex]\( z^2 + \square = 0 \)[/tex] with [tex]\( z = -7 \)[/tex] as one of the solutions, follow these steps:
1. Start by substituting [tex]\( z = -7 \)[/tex] into the equation [tex]\( z^2 + \square = 0 \)[/tex].
[tex]\[
(-7)^2 + \square = 0
\][/tex]
2. Calculate [tex]\( (-7)^2 \)[/tex]:
[tex]\[
(-7)^2 = 49
\][/tex]
3. Now the equation becomes:
[tex]\[
49 + \square = 0
\][/tex]
4. Isolate the missing number ([tex]\(\square\)[/tex]):
[tex]\[
\square = -49
\][/tex]
So, the missing number in the equation is [tex]\(-49\)[/tex].
Next, let's find the two solutions to the equation [tex]\( z^2 - 49 = 0 \)[/tex]:
1. Write the equation as:
[tex]\[
z^2 - 49 = 0
\][/tex]
2. Factor the equation (since it is a difference of squares):
[tex]\[
(z + 7)(z - 7) = 0
\][/tex]
3. Set each factor equal to zero and solve for [tex]\( z \)[/tex]:
[tex]\[
z + 7 = 0 \implies z = -7
\][/tex]
[tex]\[
z - 7 = 0 \implies z = 7
\][/tex]
Thus, the two solutions to the equation [tex]\( z^2 - 49 = 0 \)[/tex] are [tex]\( z = -7 \)[/tex] and [tex]\( z = 7 \)[/tex].
Therefore, the solutions are:
[tex]\[
z = -7 \text{ and } z = 7
\][/tex]
These are expressed in simplified, rationalized form.
