# `tan^2 x = 3 tan x` . Solve for x.

lemjay | High School Teacher | (Level 3) Senior Educator

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`tan^2 x=3tanx`

To start, subtract both sides by 3tanx.

`tan^2x-3tanx=0`

Factor left side.

`tanx(tanx-3) = 0`

Set each factor to zero and solve for x.

>>  `tan x = 0`

`x= tan^(-1)` `(0)`

Note that tangent function is zero at the horizontal axis of unit circle chart. Values of x are:

`x= 0` degrees ,     `x=180` degrees    and     `x=360` degrees

>> `tanx - 3=0`

`tanx=3`

`x=tan^(-1)` `(3)`

Also, take note that tangent function is positive at  the first and third quadrant of unit circle chart. So, values of x are:

`x= 71.6` degrees    and      `x=251.6` degrees

Since there is no indicated interval for values of x, we should consider the general solution of a tangent function.

The set of all solutions to  `tan x = y`  is  `x = tan^(-1)y + 180k` ,                where k is any integer.

Hence, the general solutions of the equation   `tan^x=3tanx`      are:

`x_1= 180k`  degrees       and        `x_2= 71.6 + 180k` degrees.

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