Sure! Let's go through the problem step by step to find the missing number and the two solutions to the equation.
First, we are given the equation:
[tex]\[ p^2 + \square = 0 \][/tex]
And we know that [tex]\( p = -\sqrt{3} \)[/tex] is a solution. We need to substitute [tex]\( p = -\sqrt{3} \)[/tex] into the equation to find the missing number.
Let's substitute [tex]\( p = -\sqrt{3} \)[/tex] into [tex]\( p^2 \)[/tex]:
[tex]\[ (-\sqrt{3})^2 \][/tex]
Calculating [tex]\( (-\sqrt{3})^2 \)[/tex]:
[tex]\[ (-\sqrt{3})^2 = (\sqrt{3})^2 = 3 \][/tex]
So now the equation looks like this:
[tex]\[ 3 + \square = 0 \][/tex]
To solve for the missing number, we need to isolate the blank:
[tex]\[ \square = -3 \][/tex]
Therefore, the missing number is:
[tex]\[ \boxed{-3} \][/tex]
Now, let's find the two solutions to the equation:
[tex]\[ p^2 - 3 = 0 \][/tex]
We can rewrite this as:
[tex]\[ p^2 = 3 \][/tex]
Taking the square root of both sides:
[tex]\[ p = \pm \sqrt{3} \][/tex]
Hence, the two solutions are:
[tex]\[ p = -\sqrt{3} \][/tex]
and
[tex]\[ p = \sqrt{3} \][/tex]
In simplified, rationalized form, the two solutions are:
[tex]\[ p = -\sqrt{3} \][/tex]
and
[tex]\[ p = \sqrt{3} \][/tex]
So, the answers are:
[tex]\[ p = -\sqrt{3} \][/tex]
[tex]\[ p = \boxed{\sqrt{3}} \][/tex]
