To solve the quadratic equation \(3x^2 - 1 = -5\), let's follow these steps:

### Step 1: Simplify the equation
First, add 5 to both sides to move all terms to one side:
\[
3x^2 - 1 + 5 = 0
\]
This simplifies to:
\[
3x^2 + 4 = 0
\]

### Step 2: Isolate \(x^2\)
Next, subtract 4 from both sides:
\[
3x^2 = -4
\]
Now, divide by 3:
\[
x^2 = -\frac{4}{3}
\]

### Step 3: Check for real solutions
Since \(x^2 = -\frac{4}{3}\) is a negative number, there are no real solutions to this equation. The solutions will be complex numbers. 

### Step 4: Solve for \(x\)
Take the square root of both sides:
\[
x = \pm \sqrt{-\frac{4}{3}} = \pm \frac{2i}{\sqrt{3}} = \pm \frac{2i\sqrt{3}}{3}
\]

Thus, the solutions are:
\[
x = \pm \frac{2i\sqrt{3}}{3}
\]
These are the two complex solutions.

Would you like more details on complex numbers or have any other questions?

Here are some related questions for practice:
1. Solve \(4x^2 + 9 = 0\).
2. Find the roots of \(x^2 + 16 = 0\).
3. What is the solution of \(2x^2 - 8 = 0\)?
4. Solve \(5x^2 + 1 = 0\) over the complex numbers.
5. Determine if \(x^2 + 25 = 0\) has real solutions.

**Tip:** When dealing with negative square roots, remember that the solutions involve imaginary numbers!

Learn how to solve the quadratic equation 3x² - 1 = -5, which results in complex solutions. This problem involves important algebraic concepts like quadratic equations and complex numbers. Suitable for students in Grades 10-12.

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