Can any kind soul help me in the below?

1. If x is obtuse and 2 tan^2 x = 5 sec x +10, find the value of tan x without using calculator.

Answer:

obtuse : 90deg < x< 180 deg

2 tan^2 x = 5 sec x + 10

2 (sec^2 x -1) = 5 sec x + 10

2 (sec^2 x -1) - 5 sec x - 10 = 0

2 sec^2 x -2 - 5 sec x - 10 = 0

(2 sec x + 3)(sec x - 4) = 0

sec x = 3/2 or sec x = 4

sec x^2 = (3/2)^2 or sec x^2 = 4^2

sec x^2 = 9/4 or sec x^2 = 16

(sec x^2) - 1 = 9/4 -1 or (sec x^2) - 1 = 16 - 1

tan x^2 = 5/4 or tan x^2 = 15

tan x = √(5/4) or tan x = √15, -√15

tan x = (√5)/2, -(√5)/2 or tan x = √15, -√15

For x to be obtuse, tan x must be in 2nd quadrant => negative

tan x = -(√5)/2 or tan x = -√15 (rejected)

My question: why is tan x = - √15 rejected?