Question

solve the quadratic equation. $x^2 + 4 = 0$. (20 points)
x = or x =
-2i 1 - 2i 2 2i -4 -2 1 + 2i 4 0

Explanation:

Step1: Rearrange the equation

We start with the quadratic equation \( x^2 + 4 = 0 \). Subtract 4 from both sides to get \( x^2=-4 \).

Step2: Take square root of both sides

Recall that for a complex number, if \( x^2 = a \), then \( x=\pm\sqrt{a} \). Here, \( a = - 4 \), and we know that \( \sqrt{-4}=\sqrt{4\times(- 1)}=\sqrt{4}\times\sqrt{-1} \). Since \( \sqrt{4} = 2 \) and \( \sqrt{-1}=i \) (where \( i \) is the imaginary unit), we have \( \sqrt{-4} = 2i \) and \( -\sqrt{-4}=-2i \). So \( x=\pm2i \), which means \( x = 2i \) or \( x=-2i \).

Answer:

\( x = 2i \) or \( x=-2i \) (corresponding to the options \( 2i \) and \( -2i \))