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-
Similar expressions
- 4*cos^4x-9*cos^2x-1=0
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4*cos^4x+9*cos^2x-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
4 2 4*cos (x) + 9*cos (x) - 1 = 0
$$\left(4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
Detail solution
Given the equation
$$\left(4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
transform
$$4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)} - 1 = 0$$
$$\left(4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$4 w^{4} + 9 w^{2} - 1 = 0$$
Do replacement
$$v = w^{2}$$
then the equation will be the:
$$4 v^{2} + 9 v - 1 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = 9$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = - \frac{9}{8} + \frac{\sqrt{97}}{8}$$
$$v_{2} = - \frac{\sqrt{97}}{8} - \frac{9}{8}$$
The final answer:
Because
$$v = w^{2}$$
then
$$w_{1} = \sqrt{v_{1}}$$
$$w_{2} = - \sqrt{v_{1}}$$
$$w_{3} = \sqrt{v_{2}}$$
$$w_{4} = - \sqrt{v_{2}}$$
then:
$$w_{1} = $$
$$\frac{0}{1} + \frac{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}}$$
$$w_{2} = $$
$$\frac{\left(-1\right) \left(- \frac{9}{8} + \frac{\sqrt{97}}{8}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}}$$
$$w_{3} = $$
$$\frac{0}{1} + \frac{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{- \frac{\sqrt{97}}{8} - \frac{9}{8}}$$
$$w_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(- \frac{\sqrt{97}}{8} - \frac{9}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{- \frac{\sqrt{97}}{8} - \frac{9}{8}}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$\left(4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
transform
$$4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)} - 1 = 0$$
$$\left(4 \cos^{4}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$4 w^{4} + 9 w^{2} - 1 = 0$$
Do replacement
$$v = w^{2}$$
then the equation will be the:
$$4 v^{2} + 9 v - 1 = 0$$
This equation is of the form
a*v^2 + b*v + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = 9$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(9)^2 - 4 * (4) * (-1) = 97
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = - \frac{9}{8} + \frac{\sqrt{97}}{8}$$
$$v_{2} = - \frac{\sqrt{97}}{8} - \frac{9}{8}$$
The final answer:
Because
$$v = w^{2}$$
then
$$w_{1} = \sqrt{v_{1}}$$
$$w_{2} = - \sqrt{v_{1}}$$
$$w_{3} = \sqrt{v_{2}}$$
$$w_{4} = - \sqrt{v_{2}}$$
then:
$$w_{1} = $$
$$\frac{0}{1} + \frac{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}}$$
$$w_{2} = $$
$$\frac{\left(-1\right) \left(- \frac{9}{8} + \frac{\sqrt{97}}{8}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}}$$
$$w_{3} = $$
$$\frac{0}{1} + \frac{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{- \frac{\sqrt{97}}{8} - \frac{9}{8}}$$
$$w_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(- \frac{\sqrt{97}}{8} - \frac{9}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{- \frac{\sqrt{97}}{8} - \frac{9}{8}}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{9}{8} + \frac{\sqrt{97}}{8} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{97}}{8} - \frac{9}{8} \right)}$$
Rapid solution
[src]
/ ______________\
| / ____ |
| / 9 \/ 97 |
x1 = - acos|- / - - + ------ | + 2*pi
\ \/ 8 8 /
$$x_{1} = - \operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi$$
/ ______________\
| / ____ |
| / 9 \/ 97 |
x2 = - acos| / - - + ------ | + 2*pi
\\/ 8 8 /
$$x_{2} = - \operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi$$
/ ______________\
| / ____ |
| / 9 \/ 97 |
x3 = acos|- / - - + ------ |
\ \/ 8 8 /
$$x_{3} = \operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
/ ______________\
| / ____ |
| / 9 \/ 97 |
x4 = acos| / - - + ------ |
\\/ 8 8 /
$$x_{4} = \operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
/ ____________\
| / ____ |
pi | / 9 \/ 97 |
x5 = -- - I*asinh| / - + ------ |
2 \\/ 8 8 /
$$x_{5} = \frac{\pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
/ ____________\
| / ____ |
pi | / 9 \/ 97 |
x6 = -- + I*asinh| / - + ------ |
2 \\/ 8 8 /
$$x_{6} = \frac{\pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
/ ____________\
| / ____ |
3*pi | / 9 \/ 97 |
x7 = ---- - I*asinh| / - + ------ |
2 \\/ 8 8 /
$$x_{7} = \frac{3 \pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
/ ____________\
| / ____ |
3*pi | / 9 \/ 97 |
x8 = ---- + I*asinh| / - + ------ |
2 \\/ 8 8 /
$$x_{8} = \frac{3 \pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}$$
x8 = 3*pi/2 + i*asinh(sqrt(9/8 + sqrt(97)/8))
Sum and product of roots
[src]
sum
/ ______________\ / ______________\ / ______________\ / ______________\ / ____________\ / ____________\ / ____________\ / ____________\
| / ____ | | / ____ | | / ____ | | / ____ | | / ____ | | / ____ | | / ____ | | / ____ |
| / 9 \/ 97 | | / 9 \/ 97 | | / 9 \/ 97 | | / 9 \/ 97 | pi | / 9 \/ 97 | pi | / 9 \/ 97 | 3*pi | / 9 \/ 97 | 3*pi | / 9 \/ 97 |
- acos|- / - - + ------ | + 2*pi + - acos| / - - + ------ | + 2*pi + acos|- / - - + ------ | + acos| / - - + ------ | + -- - I*asinh| / - + ------ | + -- + I*asinh| / - + ------ | + ---- - I*asinh| / - + ------ | + ---- + I*asinh| / - + ------ |
\ \/ 8 8 / \\/ 8 8 / \ \/ 8 8 / \\/ 8 8 / 2 \\/ 8 8 / 2 \\/ 8 8 / 2 \\/ 8 8 / 2 \\/ 8 8 /
$$\left(\left(\frac{3 \pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right) + \left(\left(\left(\operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + \left(\operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + \left(\left(- \operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi\right) + \left(- \operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi\right)\right)\right)\right) + \left(\frac{\pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right)\right) + \left(\frac{\pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right)\right)\right) + \left(\frac{3 \pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right)$$
=
8*pi
$$8 \pi$$
product
/ / ______________\ \ / / ______________\ \ / ______________\ / ______________\ / / ____________\\ / / ____________\\ / / ____________\\ / / ____________\\ | | / ____ | | | | / ____ | | | / ____ | | / ____ | | | / ____ || | | / ____ || | | / ____ || | | / ____ || | | / 9 \/ 97 | | | | / 9 \/ 97 | | | / 9 \/ 97 | | / 9 \/ 97 | |pi | / 9 \/ 97 || |pi | / 9 \/ 97 || |3*pi | / 9 \/ 97 || |3*pi | / 9 \/ 97 || |- acos|- / - - + ------ | + 2*pi|*|- acos| / - - + ------ | + 2*pi|*acos|- / - - + ------ |*acos| / - - + ------ |*|-- - I*asinh| / - + ------ ||*|-- + I*asinh| / - + ------ ||*|---- - I*asinh| / - + ------ ||*|---- + I*asinh| / - + ------ || \ \ \/ 8 8 / / \ \\/ 8 8 / / \ \/ 8 8 / \\/ 8 8 / \2 \\/ 8 8 // \2 \\/ 8 8 // \ 2 \\/ 8 8 // \ 2 \\/ 8 8 //
$$\left(- \operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} \operatorname{acos}{\left(\sqrt{- \frac{9}{8} + \frac{\sqrt{97}}{8}} \right)} \left(\frac{\pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right) \left(\frac{\pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right) \left(\frac{3 \pi}{2} - i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right) \left(\frac{3 \pi}{2} + i \operatorname{asinh}{\left(\sqrt{\frac{9}{8} + \frac{\sqrt{97}}{8}} \right)}\right)$$
=
/ / ____________\\ / / ____________\\ / / _____________ \ \ / / _____________\ \ / / ____________\\ / / ____________\\ / _____________ \ / _____________\
| | ___ / ____ || | | ___ / ____ || | | ___ / ____ | | | | ___ / ____ | | | | ___ / ____ || | | ___ / ____ || | ___ / ____ | | ___ / ____ |
| |\/ 2 *\/ 9 + \/ 97 || | |\/ 2 *\/ 9 + \/ 97 || | |-\/ 2 *\/ -9 + \/ 97 | | | |\/ 2 *\/ -9 + \/ 97 | | | |\/ 2 *\/ 9 + \/ 97 || | |\/ 2 *\/ 9 + \/ 97 || |-\/ 2 *\/ -9 + \/ 97 | |\/ 2 *\/ -9 + \/ 97 |
|pi - 2*I*asinh|---------------------||*|pi + 2*I*asinh|---------------------||*|- acos|------------------------| + 2*pi|*|- acos|----------------------| + 2*pi|*|3*pi - 2*I*asinh|---------------------||*|3*pi + 2*I*asinh|---------------------||*acos|------------------------|*acos|----------------------|
\ \ 4 // \ \ 4 // \ \ 4 / / \ \ 4 / / \ \ 4 // \ \ 4 // \ 4 / \ 4 /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16
$$\frac{\left(\pi - 2 i \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{9 + \sqrt{97}}}{4} \right)}\right) \left(\pi + 2 i \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{9 + \sqrt{97}}}{4} \right)}\right) \left(3 \pi - 2 i \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{9 + \sqrt{97}}}{4} \right)}\right) \left(3 \pi + 2 i \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{9 + \sqrt{97}}}{4} \right)}\right) \left(- \operatorname{acos}{\left(- \frac{\sqrt{2} \sqrt{-9 + \sqrt{97}}}{4} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(\frac{\sqrt{2} \sqrt{-9 + \sqrt{97}}}{4} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \frac{\sqrt{2} \sqrt{-9 + \sqrt{97}}}{4} \right)} \operatorname{acos}{\left(\frac{\sqrt{2} \sqrt{-9 + \sqrt{97}}}{4} \right)}}{16}$$
(pi - 2*i*asinh(sqrt(2)*sqrt(9 + sqrt(97))/4))*(pi + 2*i*asinh(sqrt(2)*sqrt(9 + sqrt(97))/4))*(-acos(-sqrt(2)*sqrt(-9 + sqrt(97))/4) + 2*pi)*(-acos(sqrt(2)*sqrt(-9 + sqrt(97))/4) + 2*pi)*(3*pi - 2*i*asinh(sqrt(2)*sqrt(9 + sqrt(97))/4))*(3*pi + 2*i*asinh(sqrt(2)*sqrt(9 + sqrt(97))/4))*acos(-sqrt(2)*sqrt(-9 + sqrt(97))/4)*acos(sqrt(2)*sqrt(-9 + sqrt(97))/4)/16
Numerical answer
[src]
x1 = -1.90259173789352
x2 = 11.3273696986629
x3 = -42.0797054123636
x4 = -4.38059356928607
x5 = 76.6372246018513
x6 = -70.3540392946717
x7 = 491.327454875704
x8 = 4.38059356928607
x9 = 67.8760374632792
x10 = -33.3185182737914
x11 = -93.0087786919975
x12 = -39.601703580971
x13 = 38.9381127587738
x14 = 51.504483373133
x15 = -52.1680741953302
x16 = 8.1857770450731
x17 = 74.1592227704588
x18 = 23.8937403130221
x19 = 70.3540392946717
x20 = 42.7432962345608
x21 = 89.8671860384077
x22 = 26.3717421444146
x23 = 108.716741959946
x24 = 1.90259173789352
x25 = 54.6460760267228
x26 = 23.2301494908248
x27 = 16.9469641836452
x28 = 92.3451878698003
x29 = -17.6105550058425
x30 = -49.0264815417404
x31 = 98.6283731769799
x32 = -10.6637788764657
x33 = 45.8848888881506
x34 = -92.3451878698003
x35 = 39.601703580971
x36 = -8.1857770450731
x37 = 20.088556837235
x38 = 55.30966684892
x39 = 17.6105550058425
x40 = 77.3008154240486
x41 = 64.0708539874921
x42 = 60.9292613339023
x43 = 30.1769256202017
x44 = 32.6549274515942
x45 = 86.0620025626207
x46 = -89.8671860384077
x47 = -27.0353329666119
x48 = -5.04418439148331
x49 = -57.7876686803126
x50 = -23.8937403130221
x51 = -76.6372246018513
x52 = -79.7788172554411
x53 = -121.283112574306
x54 = 42.0797054123636
x55 = 96.1503713455873
x56 = -11.3273696986629
x57 = 48.3628907195432
x58 = 82.9204099090309
x59 = -83.5840007312281
x60 = 99.2919639991771
x61 = -98.6283731769799
x62 = -64.0708539874921
x63 = -20.088556837235
x64 = -48.3628907195432
x65 = -67.8760374632792
x66 = -96.1503713455873
x67 = 83.5840007312281
x68 = 71.017630116869
x69 = -71.017630116869
x70 = -35.796520105184
x71 = -26.3717421444146
x72 = -32.6549274515942
x73 = -74.1592227704588
x74 = -45.8848888881506
x75 = -77.3008154240486
x76 = -1280.53080174894
x77 = 52.1680741953302
x78 = 33.3185182737914
x79 = -55.30966684892
x80 = 61.5928521560996
x81 = -99.2919639991771
x82 = -13.8053715300554
x83 = 10.6637788764657
x84 = -86.0620025626207
x85 = -61.5928521560996
x86 = -30.1769256202017
x87 = -54.6460760267228
x87 = -54.6460760267228