Question

use square roots to solve the equation $x^2 = -25$ over the complex numbers. select any solutions that apply.
a. $-5$
b. $-5i$
c. $-5i^2$
d. $5i$

Explanation:

Step1: Recall complex number definition

In complex numbers, \(i^2 = -1\), so we can rewrite \(-25\) as \(25i^2\) (since \(25\times(-1)= -25\)). The equation becomes \(x^2 = 25i^2\).

Step2: Take square roots of both sides

Taking the square root of both sides, we have \(x=\pm\sqrt{25i^2}\). Since \(\sqrt{25i^2}=\sqrt{25}\times\sqrt{i^2}=5|i|\) (but in complex numbers, we know \(i^2 = -1\), so \(\sqrt{i^2}=i\) in the sense of the imaginary unit), so \(x = \pm5i\). That means the solutions are \(5i\) and \(-5i\).

Answer:

B. \(-5i\), D. \(5i\)