7tg^2x+4tgx=0 (уравнение)

Найду корень уравнения: 7tg^2x+4tgx=0

Вы ввели

     2                  
7*tan (x) + 4*tan(x) = 0

$$7 \tan^{2}{\left(x \right)} + 4 \tan{\left(x \right)} = 0$$

Подробное решение

Дано уравнение
$$7 \tan^{2}{\left(x \right)} + 4 \tan{\left(x \right)} = 0$$
преобразуем
$$\left(7 \tan{\left(x \right)} + 4\right) \tan{\left(x \right)} = 0$$
$$\left(7 \tan^{2}{\left(x \right)} + 4 \tan{\left(x \right)}\right) + 0 = 0$$
Сделаем замену
$$w = \tan{\left(x \right)}$$
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.
$$a = 7$$
$$b = 4$$
$$c = 0$$
, то

D = b^2 - 4 * a * c = 

(4)^2 - 4 * (7) * (0) = 16

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

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

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

или
$$w_{1} = 0$$
Упростить
$$w_{2} = - \frac{4}{7}$$
Упростить
делаем обратную замену
$$\tan{\left(x \right)} = w$$
Дано уравнение
$$\tan{\left(x \right)} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Или
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, где n - любое целое число
подставляем w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(0 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(- \frac{4}{7} \right)}$$
$$x_{2} = \pi n - \operatorname{atan}{\left(\frac{4}{7} \right)}$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

x2 = -atan(4/7)

$$x_{2} = - \operatorname{atan}{\left(\frac{4}{7} \right)}$$

Сумма и произведение корней

сумма

0 + 0 - atan(4/7)

$$- \operatorname{atan}{\left(\frac{4}{7} \right)} + \left(0 + 0\right)$$

=

-atan(4/7)

$$- \operatorname{atan}{\left(\frac{4}{7} \right)}$$

произведение

1*0*-atan(4/7)

$$1 \cdot 0 \left(- \operatorname{atan}{\left(\frac{4}{7} \right)}\right)$$

=

0

$$0$$

Численный ответ

x1 = 18.3304098072922

x2 = -97.3893722612836

x3 = -72.2566310325652

x4 = 31.4159265358979

x5 = -34.5575191894877

x6 = 50.2654824574367

x7 = 12.0472245001126

x8 = 30.8967804216514

x9 = 62.8318530717959

x10 = -69.1150383789755

x11 = 46.6047436896004

x12 = 43.4631510360106

x13 = -28.2743338823081

x14 = -13.0855167286057

x15 = 34.0383730752412

x16 = -57.0678138788628

x17 = -65.9734457253857

x18 = -100.530964914873

x19 = 87.9645943005142

x20 = 8.90563184652286

x21 = 91.106186954104

x22 = -91.106186954104

x23 = -12.5663706143592

x24 = -37.6991118430775

x25 = -21.9911485751286

x26 = -41.3598506109138

x27 = 24.6135951144718

x28 = 59.6902604182061

x29 = 78.0206702254983

x30 = 74.8790775719085

x31 = 59.1711143039595

x32 = -6.28318530717959

x33 = 28.2743338823081

x34 = -62.8318530717959

x35 = -15.707963267949

x36 = 100.011818800627

x37 = 52.88792899678

x38 = 65.9734457253857

x39 = -94.2477796076938

x40 = -3.66073876783632

x41 = 81.1622628790881

x42 = -19.3687020357853

x43 = -56.5486677646163

x44 = -50.2654824574367

x45 = -40.8407044966673

x46 = -75.9173698004016

x47 = -31.4159265358979

x48 = 2.62244653934327

x49 = 3.14159265358979

x50 = -84.8230016469244

x51 = -78.5398163397448

x52 = -35.0766653037342

x53 = 72.2566310325652

x54 = 0.0

x55 = 62.3127069575493

x56 = 43.9822971502571

x57 = -75.398223686155

x58 = 53.4070751110265

x59 = 21.9911485751286

x60 = -43.9822971502571

x61 = -18.8495559215388

x62 = 40.8407044966673

x63 = 12.5663706143592

x64 = -3.14159265358979

x65 = 75.398223686155

x66 = 84.8230016469244

x67 = 56.0295216503698

x68 = -85.3421477611709

x69 = 18.8495559215388

x70 = 40.3215583824208

x71 = -63.3509991860424

x72 = 25.1327412287183

x73 = 6.28318530717959

x74 = 47.1238898038469

x75 = -47.1238898038469

x76 = -25.1327412287183

x77 = -9.42477796076938

x78 = 81.6814089933346

x79 = -31.9350726501445

x80 = -69.634184493222

x81 = 90.5870408398575

x82 = -59.6902604182061

x83 = -97.9085183755301

x84 = 37.6991118430775

x85 = 9.42477796076938

x86 = -53.4070751110265

x87 = -9.9439240750159

x88 = -79.0589624539914

x89 = -91.6253330683505

x90 = -53.926221225273

x91 = 68.5958922647289

x92 = -87.9645943005142

x93 = -25.6518873429649

x94 = 97.3893722612836

x95 = -47.6430359180934

x96 = 15.707963267949

x97 = 96.8702261470371

x98 = 84.3038555326779

x99 = 69.1150383789755

x100 = 94.2477796076938

x101 = -81.6814089933346

График