Question

How do you solve `6tan^2x=5tanx+sec^2x`?

Answer 1

`49^@48 + k180`
`-9^@65 + k180`

Explanation:

`6tan^2x = 5tan x + sec^2 x`
Call tan x = t and use the trig identity:
`sec^2 t = 1 + tan^2 t`
`6t^2 = 5t + 1 + t^2`
Solve this quadratic equation for t by the improved quadratic formula (Socratic Search)
`5t^2 - 5t - 1 = 0`
`D = d^2 = b^2 - 4ac = 25 + 20 = 45` --> `d = +- 3sqrt5`
There are 2 real roots:
`t = -b/(2a) +- d/(2a) =`
`t = 5/10 +- (3sqrt5)/10 = 1/2 +- (3sqrt5)/10`
t1 = 0.5 + 0.67 = 1.17, and t2 = 0.5 - 0.67 = - 0.17.
Calculator -->
a tan x1 = 1.17 --> `x1 = 49^@48 + k180`
b. tan x2 = -0.17 --> `x2 = -9^@65 + k180`