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- Equation:
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Equation 2x^2+6x=0
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Equation x^2-14x+40=0
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Identical expressions
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Similar expressions
- tg*x^2-tg*x+2=0
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tg*x^2-tg*x-2=0 equation
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The solution
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[src]
2 tan (x) - tan(x) - 2 = 0
$$\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)}\right) - 2 = 0$$
Detail solution
Given the equation
$$\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)}\right) - 2 = 0$$
transform
$$\tan^{2}{\left(x \right)} - \tan{\left(x \right)} - 2 = 0$$
$$\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)}\right) - 2 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 2$$
$$w_{2} = -1$$
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:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
$$\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)}\right) - 2 = 0$$
transform
$$\tan^{2}{\left(x \right)} - \tan{\left(x \right)} - 2 = 0$$
$$\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)}\right) - 2 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-2) = 9
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 2$$
$$w_{2} = -1$$
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:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
Rapid solution
[src]
-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
x2 = atan(2)
$$x_{2} = \operatorname{atan}{\left(2 \right)}$$
x2 = atan(2)
Sum and product of roots
[src]
sum
pi - -- + atan(2) 4
$$- \frac{\pi}{4} + \operatorname{atan}{\left(2 \right)}$$
=
pi - -- + atan(2) 4
$$- \frac{\pi}{4} + \operatorname{atan}{\left(2 \right)}$$
product
-pi ----*atan(2) 4
$$- \frac{\pi}{4} \operatorname{atan}{\left(2 \right)}$$
=
-pi*atan(2)
------------
4
$$- \frac{\pi \operatorname{atan}{\left(2 \right)}}{4}$$
-pi*atan(2)/4
Numerical answer
[src]
x1 = -79.3252145031423
x2 = -27.167185164514
x3 = -85.6083998103219
x4 = 55.7632696012188
x5 = -68.0078896611814
x6 = 13.6735193321533
x7 = 84.037603483527
x8 = 26.2398899465124
x9 = -32.2013246992954
x10 = -12207.1219031321
x11 = 24.3473430653209
x12 = 52.621676947629
x13 = 35.6646679072818
x14 = 46.3384916404494
x15 = -8.31762924297529
x16 = -2.0344439357957
x17 = 85.9301503647185
x18 = 4.24874137138388
x19 = -25.9181393921158
x20 = 1001.38265833175
x21 = -76.1836218495525
x22 = -19.6349540849362
x23 = 68.329640215578
x24 = 96.6039740978861
x25 = -30.3087778181038
x26 = -16.4933614313464
x27 = -91.8915851175014
x28 = -651.095077456485
x29 = -13.3517687777566
x30 = -61.7247043540018
x31 = -82.4668071567321
x32 = -60.4756585816035
x33 = 77.7544181763474
x34 = -89.9990382363099
x35 = -47.9092879672443
x36 = 98.4965209790777
x37 = 48.231038521641
x38 = -52.2999263932324
x39 = 92.2133356718981
x40 = 5.49778714378214
x41 = 33.7721210260903
x42 = -98.174770424681
x43 = 80.8960108299372
x44 = -46.0167410860528
x45 = 41.9478532144614
x46 = -57.3340659280137
x47 = 8.63937979737193
x48 = 40.0553063332699
x49 = -69.9004365423729
x50 = -74.2910749683609
x51 = -54.1924732744239
x52 = 90.3207887907066
x53 = 74.6128255227576
x54 = 62.0464549083984
x55 = 27.4889357189107
x56 = -96.2822235434895
x57 = 63.93900178959
x58 = -24.0255925109243
x59 = 70.2221870967695
x60 = -17.7424072037447
x61 = -83.7158529291303
x62 = -3.92699081698724
x63 = -51.0508806208341
x64 = -478.307481509046
x65 = 99.7455667514759
x66 = 49.4800842940392
x67 = -63.6172512351933
x68 = -39.7335557788732
x69 = 2.35619449019234
x70 = 11.7809724509617
x71 = -10.2101761241668
x72 = 19.9567046393328
x73 = 18.0641577581413
x74 = -41.6261026600648
x75 = 30.6305283725005
x76 = -38.484510006475
x77 = -35.3429173528852
x78 = -858.440192593411
x78 = -858.440192593411