Fill in the missing number so that [tex]q=-i \sqrt{2}[/tex] is a solution of the equation.

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

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

[tex]
q=-i \sqrt{2} \text { and } q=
[/tex]

[tex]
\boxed{i \sqrt{2}}
[/tex]

[tex]
\text{Submit}
[/tex]



Answer :

Let's start by analyzing the given equation:
[tex]\[ q^2 + x = 0 \][/tex]
We are given that [tex]\( q = -i \sqrt{2} \)[/tex] is a solution to this equation. First, let's substitute [tex]\( q = -i \sqrt{2} \)[/tex] into the equation and solve for [tex]\( x \)[/tex]:

[tex]\[ (-i \sqrt{2})^2 + x = 0 \][/tex]

We need to calculate [tex]\( (-i \sqrt{2})^2 \)[/tex]:

[tex]\[ (-i \sqrt{2})^2 = (-i \sqrt{2}) \cdot (-i \sqrt{2}) = (-i) \cdot (-i) \cdot (\sqrt{2} \cdot \sqrt{2}) \][/tex]

Since [tex]\( (-i) \cdot (-i) = i^2 = -1 \)[/tex] and [tex]\( \sqrt{2} \cdot \sqrt{2} = 2 \)[/tex], we have:

[tex]\[ (-i \sqrt{2})^2 = (-1) \cdot 2 = -2 \][/tex]

So the equation becomes:

[tex]\[ -2 + x = 0 \][/tex]

Solving for [tex]\( x \)[/tex] gives us:

[tex]\[ x = 2 \][/tex]

Thus, the given equation is:

[tex]\[ q^2 - 2 = 0 \][/tex]

Next, let’s solve this quadratic equation for [tex]\( q \)[/tex]. We can rewrite it in standard form:

[tex]\[ q^2 = 2 \][/tex]

To find the solutions, take the square root of both sides:

[tex]\[ q = \pm \sqrt{2} \][/tex]

Since we need to account for the imaginary unit [tex]\( i \)[/tex], the two solutions will be:

[tex]\[ q = \pm i \sqrt{2} \][/tex]

So, we have two solutions:

[tex]\[ q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2} \][/tex]

Therefore, the two solutions to this equation are:
[tex]\[ q = -i \sqrt{2} \text{ and } q = i \sqrt{2} \][/tex]

Expressed in a simplified, rationalized form:

[tex]\[ q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2} \][/tex]