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tan(x)^(3)-tan(x)^(2)-3*tan(x)+3=0 equation

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Numerical solution:

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The solution

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   3         2                      
tan (x) - tan (x) - 3*tan(x) + 3 = 0
$$\left(\left(\tan^{3}{\left(x \right)} - \tan^{2}{\left(x \right)}\right) - 3 \tan{\left(x \right)}\right) + 3 = 0$$
Detail solution
Given the equation
$$\left(\left(\tan^{3}{\left(x \right)} - \tan^{2}{\left(x \right)}\right) - 3 \tan{\left(x \right)}\right) + 3 = 0$$
transform
$$\tan^{3}{\left(x \right)} - \tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)} + 3 = 0$$
$$\left(\left(\tan^{3}{\left(x \right)} - \tan^{2}{\left(x \right)}\right) - 3 \tan{\left(x \right)}\right) + 3 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
Given the equation:
$$w^{3} - w^{2} - 3 w + 3 = 0$$
transform
$$\left(- 3 w + \left(\left(- w^{2} + \left(w^{3} - 1\right)\right) + 1\right)\right) + 3 = 0$$
or
$$\left(- 3 w + \left(\left(- w^{2} + \left(w^{3} - 1^{3}\right)\right) + 1^{2}\right)\right) + 3 = 0$$
$$- 3 \left(w - 1\right) + \left(- (w^{2} - 1^{2}) + \left(w^{3} - 1^{3}\right)\right) = 0$$
$$- 3 \left(w - 1\right) + \left(- (w - 1) \left(w + 1\right) + \left(w - 1\right) \left(\left(w^{2} + w\right) + 1^{2}\right)\right) = 0$$
Take common factor -1 + w from the equation
we get:
$$\left(w - 1\right) \left(\left(- (w + 1) + \left(\left(w^{2} + w\right) + 1^{2}\right)\right) - 3\right) = 0$$
or
$$\left(w - 1\right) \left(w^{2} - 3\right) = 0$$
then:
$$w_{1} = 1$$
and also
we get the equation
$$w^{2} - 3 = 0$$
This equation is of the form
a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = -3$$
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-3) = 12

Because D > 0, then the equation has two roots.
w2 = (-b + sqrt(D)) / (2*a)

w3 = (-b - sqrt(D)) / (2*a)

or
$$w_{2} = \sqrt{3}$$
$$w_{3} = - \sqrt{3}$$
The final answer for tan(x)^3 - tan(x)^2 - 3*tan(x) + 3 = 0:
$$w_{1} = 1$$
$$w_{2} = \sqrt{3}$$
$$w_{3} = - \sqrt{3}$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(\sqrt{3} \right)}$$
$$x_{2} = \pi n + \frac{\pi}{3}$$
$$x_{3} = \pi n + \operatorname{atan}{\left(w_{3} \right)}$$
$$x_{3} = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
$$x_{3} = \pi n - \frac{\pi}{3}$$
The graph
Sum and product of roots [src]
sum
  pi   pi   pi
- -- + -- + --
  3    4    3 
$$\left(- \frac{\pi}{3} + \frac{\pi}{4}\right) + \frac{\pi}{3}$$
=
pi
--
4 
$$\frac{\pi}{4}$$
product
-pi  pi pi
----*--*--
 3   4  3 
$$\frac{\pi}{3} \cdot - \frac{\pi}{3} \frac{\pi}{4}$$
=
   3 
-pi  
-----
  36 
$$- \frac{\pi^{3}}{36}$$
-pi^3/36
Rapid solution [src]
     -pi 
x1 = ----
      3  
$$x_{1} = - \frac{\pi}{3}$$
     pi
x2 = --
     4 
$$x_{2} = \frac{\pi}{4}$$
     pi
x3 = --
     3 
$$x_{3} = \frac{\pi}{3}$$
x3 = pi/3
Numerical answer [src]
x1 = 90.0589894029074
x2 = -46.3384916404494
x3 = 38.484510006475
x4 = -87.1791961371168
x5 = -90.0589894029074
x6 = 74.3510261349584
x7 = 32.2013246992954
x8 = -85.870199198121
x9 = 7.06858347057703
x10 = 60.4756585816035
x11 = 79.5870138909414
x12 = 70.162235930172
x13 = -24.0855436775217
x14 = -79.5870138909414
x15 = 80.634211442138
x16 = 47.9092879672443
x17 = 35.3429173528852
x18 = 46.0766922526503
x19 = 327.511034136736
x20 = 524175.51964962
x21 = 30.3687289847013
x22 = 54.1924732744239
x23 = -52.3598775598299
x24 = -65.1880475619882
x25 = -74.3510261349584
x26 = 85.870199198121
x27 = 474538.355722902
x28 = -21.2057504117311
x29 = 41.8879020478639
x30 = 73.3038285837618
x31 = -140.586271248143
x32 = 16.4933614313464
x33 = 25.9181393921158
x34 = -33.7721210260903
x35 = -41.8879020478639
x36 = 63.8790506229925
x37 = -19.8967534727354
x38 = 14.6607657167524
x39 = 44.7676953136546
x40 = 286.670329640069
x41 = 22.776546738526
x42 = 51.3126800086333
x43 = -96.342174710087
x44 = 95.2949771588904
x45 = -49.4800842940392
x46 = -5.49778714378214
x47 = 10.2101761241668
x48 = 58.6430628670095
x49 = 76.1836218495525
x50 = -224.100275956072
x51 = -5.23598775598299
x52 = -68.0678408277789
x53 = -55.7632696012188
x54 = -93.4623814442964
x55 = -13.6135681655558
x56 = -40.0553063332699
x57 = 0.785398163397448
x58 = -68.329640215578
x59 = -99.7455667514759
x60 = 66.7588438887831
x61 = 13.6135681655558
x62 = 96.342174710087
x63 = 95.0331777710912
x64 = 52.3598775598299
x65 = -77.7544181763474
x66 = 3.92699081698724
x67 = -30.3687289847013
x68 = -18.0641577581413
x69 = -84.037603483527
x70 = -8.63937979737193
x71 = 36.6519142918809
x72 = -46.0766922526503
x73 = -35.6047167406843
x74 = -11.7809724509617
x75 = 108.908545324446
x76 = 88.7499924639117
x77 = -8.37758040957278
x78 = 55.5014702134197
x79 = -43.1968989868597
x80 = -57.5958653158129
x81 = 57.3340659280137
x82 = -71.4712328691678
x83 = -27.4889357189107
x84 = 24.0855436775217
x85 = 82.4668071567321
x86 = -2.0943951023932
x87 = 98.174770424681
x88 = 4434.88162931759
x89 = -63.8790506229925
x90 = 2.0943951023932
x91 = 68.0678408277789
x92 = 19.8967534727354
x93 = 8.37758040957278
x93 = 8.37758040957278