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How to use it?
- Equation:
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Equation (2log3*2log3)*(8sinx−√3)−7log3*(8sinx−√3)+6=0
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Identical expressions
- (2log three *2log three)*(8sinx−√3)−7log3*(8sinx−√3)+ six = zero
- (2 logarithm of 3 multiply by 2 logarithm of 3) multiply by (8 sinus of x−√3)−7 logarithm of 3 multiply by (8 sinus of x−√3) plus 6 equally 0
- (2 logarithm of three multiply by 2 logarithm of three) multiply by (8 sinus of x−√3)−7 logarithm of 3 multiply by (8 sinus of x−√3) plus six equally zero
- (2log32log3)(8sinx−√3)−7log3(8sinx−√3)+6=0
- 2log32log38sinx−√3−7log38sinx−√3+6=0
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Similar expressions
- (2log3*2log3)*(8sinx−√3)−7log3*(8sinx−√3)-6=0
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What you mean?
- -7*(8*sin(x) - sqrt(3))*log(3) + 2*(8*sin(x) - sqrt(3))*log(3)*log(3)^2 + 6 = 0
You entered:
(2log3*2log3)*(8sinx−√3)−7log3*(8sinx−√3)+6=0
What you mean?
(2log3*2log3)*(8sinx−√3)−7log3*(8sinx−√3)+6=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ ___\ / ___\ 2*log(3)*2*log(3)*\8*sin(x) - \/ 3 / - 7*log(3)*\8*sin(x) - \/ 3 / + 6 = 0
$$- 7 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)} + 2 \log{\left(3 \right)} 2 \log{\left(3 \right)} \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6 = 0$$
Detail solution
Given the equation
$$- 7 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)} + 2 \log{\left(3 \right)} 2 \log{\left(3 \right)} \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6 = 0$$
transform
$$\left(- 8 \sin{\left(x \right)} + \sqrt{3}\right) \log{\left(2187 \right)} + 4 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)}^{2} + 6 = 0$$
$$\left(- 7 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)} + 2 \log{\left(3 \right)} 2 \log{\left(3 \right)} \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6\right) + 0 = 0$$
Do replacement
$$w = \log{\left(3 \right)}$$
Expand the expression in the equation
$$4 w^{2} \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) - 7 w \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6 = 0$$
We get the quadratic equation
$$32 w^{2} \sin{\left(x \right)} - 4 \sqrt{3} w^{2} - 56 w \sin{\left(x \right)} + 7 \sqrt{3} w + 6 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 32 \sin{\left(x \right)} - 4 \sqrt{3}$$
$$b = - 56 \sin{\left(x \right)} + 7 \sqrt{3}$$
$$c = 6$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 4 \cdot \left(32 \sin{\left(x \right)} - 4 \sqrt{3}\right) 6 = \left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}$$
The equation has two roots.
$$w_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$w_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$w_{1} = \frac{\sqrt{\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}} + 56 \sin{\left(x \right)} - 7 \sqrt{3}}{64 \sin{\left(x \right)} - 8 \sqrt{3}}$$
Simplify
$$w_{2} = \frac{- \sqrt{\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}} + 56 \sin{\left(x \right)} - 7 \sqrt{3}}{64 \sin{\left(x \right)} - 8 \sqrt{3}}$$
Simplify
do backward replacement
$$\log{\left(3 \right)} = w$$
substitute w:
$$- 7 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)} + 2 \log{\left(3 \right)} 2 \log{\left(3 \right)} \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6 = 0$$
transform
$$\left(- 8 \sin{\left(x \right)} + \sqrt{3}\right) \log{\left(2187 \right)} + 4 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)}^{2} + 6 = 0$$
$$\left(- 7 \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) \log{\left(3 \right)} + 2 \log{\left(3 \right)} 2 \log{\left(3 \right)} \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6\right) + 0 = 0$$
Do replacement
$$w = \log{\left(3 \right)}$$
Expand the expression in the equation
$$4 w^{2} \cdot \left(8 \sin{\left(x \right)} - \sqrt{3}\right) - 7 w \left(8 \sin{\left(x \right)} - \sqrt{3}\right) + 6 = 0$$
We get the quadratic equation
$$32 w^{2} \sin{\left(x \right)} - 4 \sqrt{3} w^{2} - 56 w \sin{\left(x \right)} + 7 \sqrt{3} w + 6 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 32 \sin{\left(x \right)} - 4 \sqrt{3}$$
$$b = - 56 \sin{\left(x \right)} + 7 \sqrt{3}$$
$$c = 6$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 4 \cdot \left(32 \sin{\left(x \right)} - 4 \sqrt{3}\right) 6 = \left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}$$
The equation has two roots.
$$w_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$w_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$w_{1} = \frac{\sqrt{\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}} + 56 \sin{\left(x \right)} - 7 \sqrt{3}}{64 \sin{\left(x \right)} - 8 \sqrt{3}}$$
Simplify
$$w_{2} = \frac{- \sqrt{\left(- 56 \sin{\left(x \right)} + 7 \sqrt{3}\right)^{2} - 768 \sin{\left(x \right)} + 96 \sqrt{3}} + 56 \sin{\left(x \right)} - 7 \sqrt{3}}{64 \sin{\left(x \right)} - 8 \sqrt{3}}$$
Simplify
do backward replacement
$$\log{\left(3 \right)} = w$$
substitute w:
Rapid solution
[src]
/ / ___\\
| ___ 2 | \/ 3 ||
|6 - 4*\/ 3 *log (3) + log\2187 /|
x_1 = pi - asin|------------------------------------|
\ 8*(7 - log(81))*log(3) /
$$x_{1} = - \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)} + \pi$$
/ / ___\\
| ___ 2 | \/ 3 ||
|6 - 4*\/ 3 *log (3) + log\2187 /|
x_2 = asin|------------------------------------|
\ 8*(7 - log(81))*log(3) /
$$x_{2} = \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)}$$
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\
| ___ 2 | \/ 3 || | ___ 2 | \/ 3 ||
|6 - 4*\/ 3 *log (3) + log\2187 /| |6 - 4*\/ 3 *log (3) + log\2187 /|
pi - asin|------------------------------------| + asin|------------------------------------|
\ 8*(7 - log(81))*log(3) / \ 8*(7 - log(81))*log(3) /
$$\left(- \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)} + \pi\right) + \left(\operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)}\right)$$
=
pi
$$\pi$$
product
/ / ___\\ / / ___\\
| ___ 2 | \/ 3 || | ___ 2 | \/ 3 ||
|6 - 4*\/ 3 *log (3) + log\2187 /| |6 - 4*\/ 3 *log (3) + log\2187 /|
pi - asin|------------------------------------| * asin|------------------------------------|
\ 8*(7 - log(81))*log(3) / \ 8*(7 - log(81))*log(3) /
$$\left(- \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)} + \pi\right) * \left(\operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)}\right)$$
=
/ / / ___\\\ / / ___\\ | | ___ 2 | \/ 3 ||| | ___ 2 | \/ 3 || | |6 - 4*\/ 3 *log (3) + log\2187 /|| |6 - 4*\/ 3 *log (3) + log\2187 /| |pi - asin|------------------------------------||*asin|------------------------------------| \ \ 8*(7 - log(81))*log(3) // \ 8*(7 - log(81))*log(3) /
$$\left(- \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)} + \pi\right) \operatorname{asin}{\left(\frac{- 4 \sqrt{3} \log{\left(3 \right)}^{2} + 6 + \log{\left(2187^{\sqrt{3}} \right)}}{8 \cdot \left(- \log{\left(81 \right)} + 7\right) \log{\left(3 \right)}} \right)}$$
Numerical answer
[src]
x1 = -43.4833332895408
x2 = -37.2001479823612
x3 = -5.78422144646331
x4 = -9.92374182148566
x5 = 132.445855311488
x6 = -97.8883361219999
x7 = -3.64055651430608
x8 = -53.9060389717428
x9 = -204.702486344053
x10 = 109.456779014926
x11 = -35.056483050204
x12 = -56.0497039039
x13 = -79.0387802004611
x14 = -22.4901124358448
x15 = -864.436943597909
x16 = 63.3308169325121
x17 = -68.6160745182592
x18 = 57.0476316253326
x19 = -28.7732977430244
x20 = -12.0674067536429
x21 = -41.3396683573836
x22 = 65.4744818646694
x23 = -30.9169626751817
x24 = -93.7488157469775
x25 = 44.4812610109734
x26 = -198.419301036873
x27 = -60.1892242789224
x28 = 59.1912965574898
x29 = -16.2069271286652
x30 = 207.844078997643
x31 = 21.4921847144123
x32 = -49.7665185967204
x33 = 52.9081112503102
x34 = -91.6051508148203
x35 = 40.341740635951
x36 = -81.1824451326183
x37 = 0.498963860716282
x38 = -18.3505920608225
x39 = 84.3240377862081
x40 = -100.032001054157
x41 = 8.9258141000531
x42 = -72.7555948932815
x43 = 19.348519782255
x44 = 96.8904084005673
x45 = 94.7467434684101
x46 = 78.0408524790285
x47 = 31.9148903966142
x48 = 46.6249259431306
x49 = -62.3328892110796
x50 = 75.8971875468713
x51 = 101.02992877559
x52 = 25.6317050894346
x53 = 13.0653344750755
x54 = 50.764446318153
x55 = -47.6228536645632
x56 = 38.1980757037938
x57 = 90.6072230933877
x58 = 15.2089994072327
x59 = 6.78214916789587
x60 = -66.4724095861019
x61 = -74.8992598254388
x62 = 69.6140022396917
x63 = -87.4656304397979
x64 = -24.6337773680021
x65 = -85.3219655076407
x66 = 88.4635581612305
x67 = 34.0585553287714
x68 = 82.1803728540509
x69 = 71.757667171849
x70 = 27.7753700215919
x71 = 2.64262879287351
x71 = 2.64262879287351
The graph