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7ctg^2(x)+2ctg(x)-5=0 equation

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Numerical solution:

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The solution

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     2                      
7*cot (x) + 2*cot(x) - 5 = 0
(7cot⁡2(x)+2cot⁡(x))−5=0\left(7 \cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)}\right) - 5 = 0
Simplify
Detail solution
Given the equation
(7cot⁡2(x)+2cot⁡(x))−5=0\left(7 \cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)}\right) - 5 = 0
transform
7cot⁡2(x)+2cot⁡(x)−5=07 \cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)} - 5 = 0
(7cot⁡2(x)+2cot⁡(x))−5=0\left(7 \cot^{2}{\left(x \right)} + 2 \cot{\left(x \right)}\right) - 5 = 0
Do replacement
w=cot⁡(x)w = \cot{\left(x \right)}
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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=7a = 7
b=2b = 2
c=−5c = -5
, then
D = b^2 - 4 * a * c = 

(2)^2 - 4 * (7) * (-5) = 144

Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
w1=57w_{1} = \frac{5}{7}
w2=−1w_{2} = -1
do backward replacement
cot⁡(x)=w\cot{\left(x \right)} = w
substitute w:
The graph
0-80-60-40-2020406080-100100-10000001000000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
     -pi 
x1 = ----
      4
x1=−π4x_{1} = - \frac{\pi}{4} 
x2 = acot(5/7)
x2=acot⁡(57)x_{2} = \operatorname{acot}{\left(\frac{5}{7} \right)} 
x2 = acot(5/7)
Sum and product of roots [src] 
sum
  pi            
- -- + acot(5/7)
  4
−π4+acot⁡(57)- \frac{\pi}{4} + \operatorname{acot}{\left(\frac{5}{7} \right)} 
=
  pi            
- -- + acot(5/7)
  4
−π4+acot⁡(57)- \frac{\pi}{4} + \operatorname{acot}{\left(\frac{5}{7} \right)} 
product
-pi           
----*acot(5/7)
 4
−π4acot⁡(57)- \frac{\pi}{4} \operatorname{acot}{\left(\frac{5}{7} \right)} 
=
-pi*acot(5/7) 
--------------
      4
−πacot⁡(57)4- \frac{\pi \operatorname{acot}{\left(\frac{5}{7} \right)}}{4} 
-pi*acot(5/7)/4
Numerical answer [src] 
x1 = -43.031750309445
x2 = -41.6261026600648
x3 = -33.6069723486756
x4 = 18.0641577581413
x5 = 41.7912513374794
x6 = 40.0553063332699
x7 = 88.9151411413263
x8 = -47.9092879672443
x9 = -8.4742311199573
x10 = 96.6039740978861
x11 = 70.0655852197875
x12 = 4.09213949440187
x13 = -27.3237870414961
x14 = -16.4933614313464
x15 = -3.92699081698724
x16 = 16.658510108761
x17 = -63.6172512351933
x18 = -17.8990090807267
x19 = 55.7632696012188
x20 = 82.6319558341467
x21 = -21.0406017343165
x22 = -93.2972327668817
x23 = 98.3399191020957
x24 = 93.4623814442964
x25 = -10.2101761241668
x26 = -24.1821943879063
x27 = 71.4712328691678
x28 = 44.9328439910692
x29 = -85.6083998103219
x30 = 33.7721210260903
x31 = -32.2013246992954
x32 = -52.4565282702144
x33 = -82.4668071567321
x34 = -11.6158237735471
x35 = 2.35619449019234
x36 = -49.3149356166246
x37 = 26.0832880695304
x38 = -7.06858347057703
x39 = 48.074436644659
x40 = -98.174770424681
x41 = 46.3384916404494
x42 = 63.7823999126079
x43 = 5.49778714378214
x44 = -5.33263846636751
x45 = 99.7455667514759
x46 = -91.8915851175014
x47 = -105.863603381241
x48 = 32.36647337671
x49 = -65.0228988845736
x50 = -19.6349540849362
x51 = -60.4756585816035
x52 = 27.4889357189107
x53 = -99.5804180740613
x54 = 54.3576219518386
x55 = 66.9239925661977
x56 = 84.037603483527
x57 = -76.1836218495525
x58 = -38.484510006475
x59 = 76.3487705269671
x60 = 68.329640215578
x61 = 62.0464549083984
x62 = -55.5981209238042
x63 = -25.9181393921158
x64 = -71.3060841917532
x65 = -74.447676845343
x66 = 90.3207887907066
x67 = 22.9416954159406
x68 = 11.7809724509617
x69 = -83.8724548061123
x70 = 19.8001027623508
x71 = 38.6496586838896
x72 = -2.19104581277772
x73 = -77.5892694989328
x74 = 74.6128255227576
x75 = -61.8813062309838
x76 = -54.1924732744239
x77 = -30.4653796950859
x78 = -87.0140474597021
x79 = -39.8901576558552
x80 = -46.1733429630348
x81 = -68.1644915381634
x82 = 60.6408072590181
x83 = 85.7735484877365
x84 = 92.0567337949161
x85 = 10.3753248015815
x86 = -69.9004365423729
x87 = 49.4800842940392
x88 = -90.1556401132919
x89 = 24.3473430653209
x90 = 77.7544181763474
x90 = 77.7544181763474
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