Question

Use inverse functions where needed to find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.) tan2 x + tan x − 30 = 0

Answer 1

I'm assuming you meant:

$tan^{2}x + tanx - 30 = 0$
Let $u = tan(x)$

Thus, we can rewrite it into a quadratic.

$u^{2} + u - 30 = 0$
$(u + 6)(u - 5) = 0$
$u = -6 or u = 5$

Hence, $tan(x) = -6$ or $tan(x) = 5$
$x = tan^{-1}(-6)$ or $x = tan^{-1}(5)$

Rewriting it in general form, we get:
$x = \pi \cdot n + tan^{-1}(5), n \in Z$
$x = \pi \cdot n - tan^{-1}(6), n \in Z$

From there, you can find your solutions from $[0, 2\pi)$