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4sin^2x+4sinx+1=0 equation
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The solution
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[src]
2 4*sin (x) + 4*sin(x) + 1 = 0
$$\left(4 \sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}\right) + 1 = 0$$
Detail solution
Given the equation
$$\left(4 \sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}\right) + 1 = 0$$
transform
$$\left(2 \sin{\left(x \right)} + 1\right)^{2} = 0$$
$$\left(4 \sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}\right) + 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = 4$$
$$b = 4$$
$$c = 1$$
, then
Because D = 0, then the equation has one root.
$$w_{1} = - \frac{1}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$\left(4 \sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}\right) + 1 = 0$$
transform
$$\left(2 \sin{\left(x \right)} + 1\right)^{2} = 0$$
$$\left(4 \sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}\right) + 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
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
$$a = 4$$
$$b = 4$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(4)^2 - 4 * (4) * (1) = 0
Because D = 0, then the equation has one root.
w = -b/2a = -4/2/(4)
$$w_{1} = - \frac{1}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
Rapid solution
[src]
-pi
x1 = ----
6
$$x_{1} = - \frac{\pi}{6}$$
7*pi
x2 = ----
6
$$x_{2} = \frac{7 \pi}{6}$$
x2 = 7*pi/6
Sum and product of roots
[src]
sum
pi 7*pi - -- + ---- 6 6
$$- \frac{\pi}{6} + \frac{7 \pi}{6}$$
=
pi
$$\pi$$
product
-pi 7*pi ----*---- 6 6
$$- \frac{\pi}{6} \frac{7 \pi}{6}$$
=
2 -7*pi ------ 36
$$- \frac{7 \pi^{2}}{36}$$
-7*pi^2/36
Numerical answer
[src]
x1 = -50.7890807586786
x2 = 91.6297853677709
x3 = -96.8657739694608
x4 = -75.9218226947368
x5 = -63.3554519746717
x6 = -71.7330322692614
x7 = 66.4970443424152
x8 = -84.2994031077318
x9 = 97.912971111731
x10 = 24.6091427997151
x11 = 85.3465999069222
x12 = -2.61799439084855
x13 = 49.7418838770006
x14 = 60.2138592227209
x15 = 93.7241810384308
x16 = 62.3082542596357
x17 = -90.5825886854278
x18 = -57.0722662605232
x19 = -15.1843648370597
x20 = 18.3259571002022
x21 = -84.2994027437743
x22 = 30.8923278441299
x23 = -25.6563402457721
x24 = -38.2227104461612
x25 = 53.9306739424951
x26 = 24.6091423271809
x27 = 22.5147471813518
x28 = -6.80678360224706
x29 = -46.600291004669
x30 = 43.4586987364776
x31 = -103.14895872614
x32 = 5.75958671547245
x33 = 79.0634146554119
x34 = -0.52359842329151
x35 = 91.6297858620216
x36 = -27.7507351061632
x37 = 72.7802294827634
x38 = 68.5914399218226
x39 = 97.9129711022954
x40 = -8.90117965766887
x41 = -78.0162175468738
x42 = 3.66519112244585
x43 = -57.0722660130515
x44 = 41.3643027576661
x45 = -31.9395254379247
x46 = 35.081117498718
x47 = 100.007366130059
x48 = -13.0899688636862
x49 = 85.3466002928585
x50 = 41.3643032114447
x51 = 87.4409958960889
x52 = 12.0427717680406
x53 = -88.4881935094566
x54 = 47.6474887113682
x55 = 68.5914394789999
x56 = -2.6179938899612
x57 = -1621.58540804471
x58 = 56.0250689537344
x59 = -69.6386375050078
x60 = -59.1666619967682
x61 = -46.600291538567
x62 = -38.2227107605573
x63 = -44.5058955832105
x64 = -25.6563400585787
x65 = -88.4881927432719
x66 = 74.8746249389767
x67 = -69.638637216784
x68 = 81.157810709676
x69 = 74.8746254329989
x70 = -101.054563163045
x71 = 47.6474882434188
x72 = -82.2050076079117
x73 = 28.7979323227871
x74 = 37.1755135538755
x75 = -63.3554514537161
x76 = -40.3171059947486
x77 = -19.3731543255808
x78 = -52.8834768137338
x79 = -34.0339203814772
x80 = 9.94837678138906
x81 = -536.68874510218
x82 = -94.7713779154064
x83 = 30.8923282850119
x84 = -65.4498471406844
x85 = -40.3171055915573
x86 = -19.3731548240611
x87 = -21.4675499796822
x88 = -90.5825881284891
x89 = 3.6651915601647
x90 = -13.0899692270065
x91 = 16.2315620577098
x92 = -75.9218227484576
x92 = -75.9218227484576