To solve the equation:

$4x^2 - 2\imaginaryI = 0$

1. Isolate the term involving $x^2$: $4x^2 = 2\imaginaryI$

2. Divide both sides by 4: $x^2 = \frac{2\imaginaryI}{4} = \frac{\imaginaryI}{2}$

3. Take the square root of both sides. Remember, taking the square root of a complex number involves finding both the magnitude and angle in polar form: $x = \pm \sqrt{\frac{\imaginaryI}{2}}$

Now, to simplify the square root of a complex number:

• Express $\frac{\imaginaryI}{2}$ in polar form: $\frac{\imaginaryI}{2} = \frac{1}{2} \text{cis}\left(\frac{\pi}{2}\right)$

Taking the square root of this: $x = \pm \sqrt{\frac{1}{2}} \text{cis}\left(\frac{\pi}{4}\right)$

This simplifies to: $x = \pm \frac{1}{\sqrt{2}} \text{cis}\left(\frac{\pi}{4}\right)$

This can also be expressed in rectangular form as: $x = \pm \frac{1}{\sqrt{2}} \left(1 + \imaginaryI\right)$

Thus, the solutions are: $x = \pm \frac{1 + \imaginaryI}{\sqrt{2}}$

Would you like further details on how to handle complex numbers in polar form or an explanation of any step?

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Here are five related questions:

1. How do you convert complex numbers to polar form?
2. What is the formula for the square root of a complex number?
3. How do you find the magnitude and argument of a complex number?
4. Can you explain the relationship between rectangular and polar forms of complex numbers?
5. What is the geometric interpretation of complex numbers?

Tip: When dealing with complex numbers, expressing them in polar form can simplify operations like multiplication, division, and taking roots.