Mister Exam
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- Equation:
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Equation lnx+1-2x=0
- Equation 14n^3-(-8n^3)+(3n^3-11n^3)
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Identical expressions
- 6sin^2x–11cosx– ten = zero
- 6 sinus of squared x–11 co sinus of e of x–10 equally 0
- 6 sinus of squared x–11 co sinus of e of x– ten equally zero
- 6sin2x–11cosx–10=0
- 6sin²x–11cosx–10=0
- 6sin to the power of 2x–11cosx–10=0
- 6sin^2x–11cosx–10=O
6sin^2x–11cosx–10=0 equation
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The solution
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[src]
2 6*sin (x) - 11*cos(x) - 10 = 0
$$\left(6 \sin^{2}{\left(x \right)} - 11 \cos{\left(x \right)}\right) - 10 = 0$$
Detail solution
Given the equation
$$\left(6 \sin^{2}{\left(x \right)} - 11 \cos{\left(x \right)}\right) - 10 = 0$$
transform
$$6 \sin^{2}{\left(x \right)} - 11 \cos{\left(x \right)} - 10 = 0$$
$$- 6 \cos^{2}{\left(x \right)} - 11 \cos{\left(x \right)} - 4 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = -6$$
$$b = -11$$
$$c = -4$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{4}{3}$$
$$w_{2} = - \frac{1}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{2 \pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{4} = \pi n - \frac{\pi}{3}$$
$$\left(6 \sin^{2}{\left(x \right)} - 11 \cos{\left(x \right)}\right) - 10 = 0$$
transform
$$6 \sin^{2}{\left(x \right)} - 11 \cos{\left(x \right)} - 10 = 0$$
$$- 6 \cos^{2}{\left(x \right)} - 11 \cos{\left(x \right)} - 4 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = -6$$
$$b = -11$$
$$c = -4$$
, then
D = b^2 - 4 * a * c =
(-11)^2 - 4 * (-6) * (-4) = 25
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{4}{3}$$
$$w_{2} = - \frac{1}{2}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{2 \pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{4}{3} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{4} = \pi n - \frac{\pi}{3}$$
Rapid solution
[src]
-2*pi
x1 = -----
3
$$x_{1} = - \frac{2 \pi}{3}$$
2*pi
x2 = ----
3
$$x_{2} = \frac{2 \pi}{3}$$
/ / ___\\ / / ___\\ x3 = 2*im\atanh\\/ 7 // - 2*I*re\atanh\\/ 7 //
$$x_{3} = 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}$$
/ / ___\\ / / ___\\ x4 = - 2*im\atanh\\/ 7 // + 2*I*re\atanh\\/ 7 //
$$x_{4} = - 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}$$
x4 = -2*im(atanh(sqrt(7))) + 2*i*re(atanh(sqrt(7)))
Sum and product of roots
[src]
sum
2*pi 2*pi / / ___\\ / / ___\\ / / ___\\ / / ___\\ - ---- + ---- + 2*im\atanh\\/ 7 // - 2*I*re\atanh\\/ 7 // + - 2*im\atanh\\/ 7 // + 2*I*re\atanh\\/ 7 // 3 3
$$\left(\left(- \frac{2 \pi}{3} + \frac{2 \pi}{3}\right) + \left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}\right)\right) + \left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}\right)$$
=
0
$$0$$
product
-2*pi 2*pi / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -----*----*\2*im\atanh\\/ 7 // - 2*I*re\atanh\\/ 7 ///*\- 2*im\atanh\\/ 7 // + 2*I*re\atanh\\/ 7 /// 3 3
$$- \frac{2 \pi}{3} \frac{2 \pi}{3} \left(2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} - 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}\right) \left(- 2 \operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} + 2 i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}\right)$$
=
2
2 / / / ___\\ / / ___\\\
16*pi *\- I*re\atanh\\/ 7 // + im\atanh\\/ 7 ///
-------------------------------------------------
9
$$\frac{16 \pi^{2} \left(\operatorname{im}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)} - i \operatorname{re}{\left(\operatorname{atanh}{\left(\sqrt{7} \right)}\right)}\right)^{2}}{9}$$
16*pi^2*(-i*re(atanh(sqrt(7))) + im(atanh(sqrt(7))))^2/9
Numerical answer
[src]
x1 = -73.3038285837618
x2 = -10.471975511966
x3 = 4.18879020478639
x4 = 83.7758040957278
x5 = 58.6430628670095
x6 = -79.5870138909414
x7 = -96.342174710087
x8 = -14.6607657167524
x9 = 85.870199198121
x10 = -353.95277230445
x11 = 77.4926187885482
x12 = 8.37758040957278
x13 = -85.870199198121
x14 = -71.2094334813686
x15 = -4.18879020478639
x16 = 98.4365698124802
x17 = 52.3598775598299
x18 = 27.2271363311115
x19 = -328.820031075732
x20 = -46.0766922526503
x21 = -64.9262481741891
x22 = -33.5103216382911
x23 = -58.6430628670095
x24 = 54.4542726622231
x25 = -67.0206432765823
x26 = 20.943951023932
x27 = -20.943951023932
x28 = 16.7551608191456
x29 = -90.0589894029074
x30 = 73.3038285837618
x31 = -16.7551608191456
x32 = -41.8879020478639
x33 = -60.7374579694027
x34 = 35.6047167406843
x35 = -123.569311041199
x36 = 92.1533845053006
x37 = -27.2271363311115
x38 = -23.0383461263252
x39 = -48.1710873550435
x40 = 46.0766922526503
x41 = 64.9262481741891
x42 = 67.0206432765823
x43 = -35.6047167406843
x44 = -29.3215314335047
x45 = 23.0383461263252
x46 = 60.7374579694027
x47 = 90.0589894029074
x48 = 10.471975511966
x49 = 96.342174710087
x50 = -2.0943951023932
x51 = 41.8879020478639
x52 = 2.0943951023932
x53 = 33.5103216382911
x54 = -83.7758040957278
x55 = -569.675467850949
x56 = 39.7935069454707
x57 = 14.6607657167524
x58 = -54.4542726622231
x59 = 79.5870138909414
x60 = -35822.5338313332
x61 = -92.1533845053006
x62 = -8.37758040957278
x63 = 71.2094334813686
x64 = -52.3598775598299
x65 = -102.625360017267
x66 = -39.7935069454707
x67 = -77.4926187885482
x68 = 29.3215314335047
x69 = -7912.62469684149
x70 = 48.1710873550435
x71 = -98.4365698124802
x71 = -98.4365698124802