Mister Exam
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How to use it?
- Equation:
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Equation 11324*x%=211
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Equation 4x^3-4x=0
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Equation x-4/3+x/2=5
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Equation -x^3+4*x^2+4*x-16=0
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Identical expressions
- cosx*(cosx- one)= zero
- co sinus of e of x multiply by ( co sinus of e of x minus 1) equally 0
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- cosxcosx-1=0
- cosx*(cosx-1)=O
-
Similar expressions
- cosx*(cosx+1)=0
cosx*(cosx-1)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
cos(x)*(cos(x) - 1) = 0
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
transform
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Expand the expression in the equation
$$w \left(w - 1\right) = 0$$
We get the quadratic equation
$$w^{2} - w = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
transform
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
$$\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Expand the expression in the equation
$$w \left(w - 1\right) = 0$$
We get the quadratic equation
$$w^{2} - w = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (0) = 1
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 1$$
$$w_{2} = 0$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{2}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{3} = \pi n - \pi$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{4} = \pi n - \frac{\pi}{2}$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
pi
x2 = --
2
$$x_{2} = \frac{\pi}{2}$$
3*pi
x3 = ----
2
$$x_{3} = \frac{3 \pi}{2}$$
x4 = 2*pi
$$x_{4} = 2 \pi$$
x4 = 2*pi
Sum and product of roots
[src]
sum
pi 3*pi -- + ---- + 2*pi 2 2
$$\left(\frac{\pi}{2} + \frac{3 \pi}{2}\right) + 2 \pi$$
=
4*pi
$$4 \pi$$
product
pi 3*pi 0*--*----*2*pi 2 2
$$2 \pi \frac{3 \pi}{2} \cdot 0 \frac{\pi}{2}$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 73.8274273593601
x2 = 25.1327418085792
x3 = 87.9645943356049
x4 = -4.71238898038469
x5 = 14.1371669411541
x6 = 58.1194640914112
x7 = 10.9955742875643
x8 = 100.530964774136
x9 = 92.6769832808989
x10 = 36.1283155162826
x11 = -56.5486675907774
x12 = -7.85398163397448
x13 = -58.1194640914112
x14 = -48.6946861306418
x15 = 86.3937979737193
x16 = 51.8362787842316
x17 = -42.4115008234622
x18 = -89.5353906273091
x19 = 1.5707963267949
x20 = 69.1150379836781
x21 = 75.3982238342404
x22 = -69.115038497193
x23 = -37.6991118770355
x24 = 61.261056745001
x25 = -81.6814090377756
x26 = -70.6858347057703
x27 = 83.2522053201295
x28 = 37.6991120060109
x29 = 81.6814091609407
x30 = -36.1283155162826
x31 = 6.28318528429551
x32 = -92.6769832808989
x33 = 98.9601685880785
x34 = 94.2477796093526
x35 = -14.1371669411541
x36 = 80.1106126665397
x37 = 95.8185759344887
x38 = 50.2654824463558
x39 = 25.1327408583892
x40 = 45.553093477052
x41 = -17.2787595947439
x42 = 56.5486676180351
x43 = 4.71238898038469
x44 = 31.4159266948554
x45 = 20.4203522483337
x46 = -23.5619449019235
x47 = -100.530964736174
x48 = -51.8362787842316
x49 = -29.845130209103
x50 = -389.557489134924
x51 = 7.85398163397448
x52 = 18.8495557729205
x53 = 69.1150385134118
x54 = -26.7035375555132
x55 = -95.8185759344887
x56 = -39.2699081698724
x57 = -6.28318514935383
x58 = -62.8318534973011
x59 = -25.1327413641924
x60 = 23.5619449019235
x61 = -10.9955742875643
x62 = -86.3937979737193
x63 = -94.2477794613449
x64 = -32.9867228626928
x65 = -43.9822971746199
x66 = 12.5663704623094
x67 = -67.5442420521806
x68 = -75.3982238479311
x69 = -18.8495562409837
x70 = 76.9690200129499
x71 = -45.553093477052
x72 = 17.2787595947439
x73 = -83.2522053201295
x74 = -87.964594358935
x75 = 42.4115008234622
x76 = -76.9690200129499
x77 = -54.9778714378214
x78 = -73.8274273593601
x79 = 0.0
x80 = -1.5707963267949
x81 = 70.6858347057703
x82 = 54.9778714378214
x83 = -31.4159266930206
x84 = -12.5663704469816
x85 = 39.2699081698724
x86 = 26.7035375555132
x87 = -98.9601685880785
x88 = 48.6946861306418
x89 = 62.8318529132021
x90 = -20.4203522483337
x91 = -80.1106126665397
x92 = 43.9822971693881
x93 = 32.9867228626928
x94 = 64.4026493985908
x95 = 29.845130209103
x96 = 89.5353906273091
x97 = -50.2654823051418
x97 = -50.2654823051418