Fill in the missing number so that [tex]$z=-7$[/tex] is a solution of the equation.

[tex]
z^2 + \square = 0
[/tex]

What are the two solutions to this equation? Write your answer in simplified, rationalized form.

[tex]
z = -7 \text{ and } z = \square
[/tex]



Answer :

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.