Mister Exam
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How to use it?
- Equation:
-
Equation x^2-8*x+17=0
-
Equation 2-3(7+2x)=11
-
Equation x^2-2x+1=0
-
Equation 7+8x=-2x-5
- Express {x} in terms of y where:
- 7*x+19*y=12
- 17*x+20*y=-12
- 18*x-11*y=2
- 2*x-8*y=4
-
Identical expressions
- 5tg^2x+11tgx− twelve = zero .
- 5tg squared x plus 11tgx−12 equally 0.
- 5tg squared x plus 11tgx− twelve equally zero .
- 5tg2x+11tgx−12=0.
- 5tg²x+11tgx−12=0.
- 5tg to the power of 2x+11tgx−12=0.
- 5tg^2x+11tgx−12=O.
-
Similar expressions
- 5tg^2x-11tgx−12=0.
5tg^2x+11tgx−12=0. equation
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The solution
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[src]
2 5*tan (x) + 11*tan(x) - 12 = 0
$$5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12 = 0$$
Detail solution
Given the equation
$$5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12 = 0$$
transform
$$5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12 = 0$$
$$\left(5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12\right) + 0 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
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
$$a = 5$$
$$b = 11$$
$$c = -12$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{4}{5}$$
Simplify
$$w_{2} = -3$$
Simplify
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{4}{5} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{4}{5} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-3 \right)}$$
$$x_{2} = \pi n - \operatorname{atan}{\left(3 \right)}$$
$$5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12 = 0$$
transform
$$5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12 = 0$$
$$\left(5 \tan^{2}{\left(x \right)} + 11 \tan{\left(x \right)} - 12\right) + 0 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
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
$$a = 5$$
$$b = 11$$
$$c = -12$$
, then
D = b^2 - 4 * a * c =
(11)^2 - 4 * (5) * (-12) = 361
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{4}{5}$$
Simplify
$$w_{2} = -3$$
Simplify
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{4}{5} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{4}{5} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-3 \right)}$$
$$x_{2} = \pi n - \operatorname{atan}{\left(3 \right)}$$
Rapid solution
[src]
x1 = atan(4/5)
$$x_{1} = \operatorname{atan}{\left(\frac{4}{5} \right)}$$
x2 = -atan(3)
$$x_{2} = - \operatorname{atan}{\left(3 \right)}$$
Sum and product of roots
[src]
sum
0 + atan(4/5) - atan(3)
$$- \operatorname{atan}{\left(3 \right)} + \left(0 + \operatorname{atan}{\left(\frac{4}{5} \right)}\right)$$
=
-atan(3) + atan(4/5)
$$- \operatorname{atan}{\left(3 \right)} + \operatorname{atan}{\left(\frac{4}{5} \right)}$$
product
1*atan(4/5)*-atan(3)
$$1 \operatorname{atan}{\left(\frac{4}{5} \right)} \left(- \operatorname{atan}{\left(3 \right)}\right)$$
=
-atan(3)*atan(4/5)
$$- \operatorname{atan}{\left(\frac{4}{5} \right)} \operatorname{atan}{\left(3 \right)}$$
-atan(3)*atan(4/5)
Numerical answer
[src]
x1 = -65.2987047831621
x2 = 82.3561499355582
x3 = -15.0332223257254
x4 = -79.7888621121431
x5 = -57.7977135370145
x6 = -13.8154163867574
x7 = -27.5995929400846
x8 = 10.0995189029929
x9 = 79.2145572819684
x10 = -11.8916296721356
x11 = 19.5242968637623
x12 = 57.2234087068398
x13 = -24.4580002864948
x14 = 88.6393352427378
x15 = -77.8650753975213
x16 = 3.81633359581335
x17 = -43.3075562080336
x18 = -37.024370900854
x19 = -84.1482607047009
x20 = -68.4402974367519
x21 = 47.7986307460705
x22 = -46.4491488616233
x23 = 91.7809278963276
x24 = 80.4323632209364
x25 = -62.1571121295723
x26 = 36.4500660706793
x27 = 69.789779321199
x28 = 74.1491779137568
x29 = 13.2411115565827
x30 = -96.71463131906
x31 = 60.3650013604296
x32 = 30.1668807634997
x33 = 96.1403264888853
x34 = 98.0641132035071
x35 = -40.1659635544438
x36 = -33.8827782472642
x37 = 28.9490748245317
x38 = -71.5818900903417
x39 = 58.4412146458078
x40 = -48.3729355762452
x41 = 22.6658895173521
x42 = -2.46685171136624
x43 = -20.098601693937
x44 = -81.0066680511111
x45 = 8.17573218837112
x46 = 35.2322601317113
x47 = -5.60844436495603
x48 = 89.8571411817058
x49 = -35.806564961886
x50 = 0.674740942223553
x51 = 52.1580293386282
x52 = 38.3738527853011
x53 = 72.9313719747888
x54 = 41.5154454388909
x55 = 25.8074821709419
x56 = 45.8748440314486
x57 = -21.316407632905
x58 = 44.6570380924807
x59 = -55.8739268223927
x60 = 16.3827042101725
x61 = -74.7234827439315
x62 = -49.5907415152131
x63 = -30.7411855936744
x64 = 76.0729646283786
x65 = 6.95792624940314
x66 = 54.08181605325
x67 = -8.75003701854583
x68 = -87.2898533582907
x69 = 67.8659926065772
x70 = 101.205705857097
x71 = -18.1748149793152
x72 = 50.9402233996602
x73 = -90.4314460118805
x74 = -99.8562239726498
x75 = -93.5730386654702
x76 = 32.0906674781215
x77 = -52.7323341688029
x78 = 94.9225205499174
x79 = -59.0155194759825
x80 = 66.6481866676092
x81 = 85.497742589148
x82 = 63.5065940140194
x82 = 63.5065940140194
The graph