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

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lemjay eNotes educator| Certified Educator

`tan^2 x=3tanx`

To start, subtract both sides by 3tanx.


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`


                 `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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