2-cos(2*x)-5*cos(x)=0

$$- cos{left (2 x right )} + 2 – 5 cos{left (x right )} = 0$$

Дано уравнение
$$- cos{left (2 x right )} + 2 – 5 cos{left (x right )} = 0$$
преобразуем
$$- 5 cos{left (x right )} – cos{left (2 x right )} + 2 = 0$$
$$- 2 cos^{2}{left (x right )} – 5 cos{left (x right )} + 3 = 0$$
Сделаем замену
$$w = cos{left (x right )}$$
Это уравнение вида

a*w^2 + b*w + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$w_{1} = frac{sqrt{D} – b}{2 a}$$
$$w_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = -2$$
$$b = -5$$
$$c = 3$$
, то

D = b^2 – 4 * a * c =

(-5)^2 – 4 * (-2) * (3) = 49

Т.к. D > 0, то уравнение имеет два корня.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = -3$$
$$w_{2} = frac{1}{2}$$
делаем обратную замену
$$cos{left (x right )} = w$$
Дано уравнение
$$cos{left (x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
Или
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (-3 right )}$$
$$x_{1} = pi n + {acos}{left (-3 right )}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (frac{1}{2} right )}$$
$$x_{2} = pi n + frac{pi}{3}$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (-3 right )}$$
$$x_{3} = pi n – pi + {acos}{left (-3 right )}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (frac{1}{2} right )}$$
$$x_{4} = pi n – frac{2 pi}{3}$$

$$x_{1} = – i log{left (frac{1}{2} – frac{sqrt{3} i}{2} right )}$$

/ ___
|1 I*/ 3 |
x2 = -I*log|- + ——-|
2 2 /

$$x_{2} = – i log{left (frac{1}{2} + frac{sqrt{3} i}{2} right )}$$

/ ___
x3 = pi – I*log3 – 2*/ 2 /

$$x_{3} = pi – i log{left (- 2 sqrt{2} + 3 right )}$$

/ ___
x4 = pi – I*log3 + 2*/ 2 /

$$x_{4} = pi – i log{left (2 sqrt{2} + 3 right )}$$

x1 = -70.1622359302000

x2 = -13.6135681656000

x3 = 11.5191730632000

x4 = -89.0117918517000

x5 = -26.1799387799000

x6 = 49.2182849062000

x7 = 86.9173967493000

x8 = 74.3510261350000

x9 = -68.0678408278000

x10 = -57.5958653158000

x11 = 109898.147010000

x12 = -63.8790506230000

x13 = -38.7463093943000

x14 = 38.7463093943000

x15 = 61.7846555206000

x16 = -86.9173967493000

x17 = -51.3126800086000

x18 = -30296.4723537000

x19 = 42.9350995991000

x20 = -61.7846555206000

x21 = 82.7286065445000

x22 = 76.4454212374000

x23 = -80.6342114421000

x24 = 13.6135681656000

x25 = -76.4454212374000

x26 = 7.33038285838000

x27 = -32.4631240871000

x28 = 63.8790506230000

x29 = 57.5958653158000

x30 = -36.6519142919000

x31 = 95.2949771589000

x32 = 36.6519142919000

x33 = 26.1799387799000

x34 = 93.2005820565000

x35 = -17.8023583703000

x36 = 19.8967534727000

x37 = 99.4837673637000

x38 = -82.7286065445000

x39 = -99.4837673637000

x40 = 32.4631240871000

x41 = 68.0678408278000

x42 = 55.5014702134000

x43 = -11.5191730632000

x44 = -1.04719755120000

x45 = 80.6342114421000

x46 = 24.0855436775000

x47 = 5.23598775598000

x48 = 51.3126800086000

x49 = -42.9350995991000

x50 = -105.766952671000

x51 = -24.0855436775000

x52 = 45.0294947015000

x53 = -49.2182849062000

x54 = -30.3687289847000

x55 = -19.8967534727000

x56 = -95.2949771589000

x57 = -93.2005820565000

x58 = -5.23598775598000

x59 = -7.33038285838000

x60 = 89.0117918517000

x61 = 1.04719755120000

x62 = 70.1622359302000

x63 = -45.0294947015000

x64 = -4503.99666770000

x65 = 17.8023583703000

x66 = -55.5014702134000

x67 = -74.3510261350000

x68 = 30.3687289847000