Question
The Existence of Imaginary Numbers Quick Check
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Question
What are the types of solutions for x in the equation x2+4=0?(1 point)
Responses
rational
rational
imaginary
imaginary
irrational
irrational
real
Answers
GPT-4o mini
To solve the equation \(x^2 + 4 = 0\), we can rearrange it:
\[
x^2 = -4
\]
To find \(x\), we take the square root of both sides. The square root of a negative number involves imaginary numbers:
\[
x = \pm \sqrt{-4} = \pm \sqrt{4}i = \pm 2i
\]
Thus, the solutions for \(x\) are \(2i\) and \(-2i\), which are imaginary numbers.
Therefore, the correct response is:
**imaginary**
\[
x^2 = -4
\]
To find \(x\), we take the square root of both sides. The square root of a negative number involves imaginary numbers:
\[
x = \pm \sqrt{-4} = \pm \sqrt{4}i = \pm 2i
\]
Thus, the solutions for \(x\) are \(2i\) and \(-2i\), which are imaginary numbers.
Therefore, the correct response is:
**imaginary**