How do you solve 4tan^2x+tanx-3=0?

Question

4tan²x+tanx-3

Answer 1

Let's temporarily substitute another variable for tan x - to avoid confusion as much as possible, we'll say

A = tan x.

Now let's rewrite the original equation in terms of A

4A² +A - 3 = 0

Now we can factor this like we would any other quadratic formula. In this case, the right factoring combination would be

(4A - 3)(A+1) = 0

So now we know the solutions to the problem are when 4A - 3 = 0 or A + 1 = 0. We can now solve for A in both cases:

4A - 3 = 0 or A +1 = 0

4A = 3 A = -1

A = ³/₄

So A = ³/₄ or A = -1

Remember that A is equal to tan x, so now we substitute that back into our solutions:

tan x = ³/₄ or tan x = -1

So x = tan^{-1 3}/₄ or x = tan^-1 -1