Solve each equation for exact solutions over the interval [0, 2π).
tan² x + 3 = 0

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Start with the given equation: \(\tan^{2} x + 3 = 0\).

Isolate the \(\tan^{2} x\) term by subtracting 3 from both sides: \(\tan^{2} x = -3\).

Recall that \(\tan^{2} x\) represents the square of the tangent function, which is always greater than or equal to zero for all real \(x\) because squaring any real number cannot produce a negative result.

Since \(\tan^{2} x = -3\) implies a negative value, recognize that there are no real values of \(x\) for which this equation holds true within the interval \([0, 2\pi)\).

Conclude that the equation has no exact solutions in the interval \([0, 2\pi)\) because the square of tangent cannot be negative.

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