Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x+970=69*32
- Equation x^4-50*x^2+49=0
-
Equation (x^2-4)^2+(x^2-3x-10)^2=0
-
Equation 3^x=7
- Express {x} in terms of y where:
- 1*x+6*y=-6
- -5*x+6*y=-8
- -2*x-9*y=10
- 11*x-12*y=11
-
Identical expressions
- 2sin^2x-11sinx+ nine = zero
- 2 sinus of squared x minus 11 sinus of x plus 9 equally 0
- 2 sinus of squared x minus 11 sinus of x plus nine equally zero
- 2sin2x-11sinx+9=0
- 2sin²x-11sinx+9=0
- 2sin to the power of 2x-11sinx+9=0
- 2sin^2x-11sinx+9=O
-
Similar expressions
- 2sin^2x+11sinx+9=0
- 2sin^2x-11sinx-9=0
2sin^2x-11sinx+9=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 2*sin (x) - 11*sin(x) + 9 = 0
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} + 9 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
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
$$a = 2$$
$$b = -11$$
$$c = 9$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{9}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
transform
$$2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} + 9 = 0$$
$$\left(2 \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)}\right) + 9 = 0$$
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
$$a = 2$$
$$b = -11$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =
(-11)^2 - 4 * (2) * (9) = 49
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{9}{2}$$
$$w_{2} = 1$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{2} = 2 \pi n + \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{9}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{\pi}{2}$$
Sum and product of roots
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sum
pi -- + pi - re(asin(9/2)) - I*im(asin(9/2)) + I*im(asin(9/2)) + re(asin(9/2)) 2
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) + \left(\frac{\pi}{2} + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)\right)$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
product
pi --*(pi - re(asin(9/2)) - I*im(asin(9/2)))*(I*im(asin(9/2)) + re(asin(9/2))) 2
$$\frac{\pi}{2} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)$$
=
-pi*(I*im(asin(9/2)) + re(asin(9/2)))*(-pi + I*im(asin(9/2)) + re(asin(9/2)))
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2
$$- \frac{\pi \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}\right)}{2}$$
-pi*(i*im(asin(9/2)) + re(asin(9/2)))*(-pi + i*im(asin(9/2)) + re(asin(9/2)))/2
Rapid solution
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pi
x1 = --
2
$$x_{1} = \frac{\pi}{2}$$
x2 = pi - re(asin(9/2)) - I*im(asin(9/2))
$$x_{2} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}$$
x3 = I*im(asin(9/2)) + re(asin(9/2))
$$x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{9}{2} \right)}\right)}$$
x3 = re(asin(9/2)) + i*im(asin(9/2))
Numerical answer
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x1 = 20.4203521467171
x2 = -61.2610565782138
x3 = -10.9955746833544
x4 = 89.5353909055217
x5 = -86.3937977315325
x6 = 7.85398174444133
x7 = -86.3937977440352
x8 = -98.9601680628786
x9 = -61.2610570080406
x10 = -48.6946857362003
x11 = -73.8274272797355
x12 = -29.845130094999
x13 = 70.6858344692885
x14 = 76.9690205715119
x15 = 1.57079658701878
x16 = -17.2787598489313
x17 = 83.2522057411799
x18 = -92.6769838313521
x19 = -54.977870904196
x20 = 64.4026493064007
x21 = 32.9867224757296
x22 = -67.5442421723436
x23 = 1.57079643579021
x24 = -10.9955737455676
x25 = -23.561945012858
x26 = 58.1194643884877
x27 = -36.1283154162869
x28 = 26.7035375458614
x29 = 70.6858346223605
x30 = 39.2699085820684
x31 = -92.6769828953542
x32 = -4.71238857705888
x33 = -42.4115005722496
x34 = 95.8185760651006
x35 = -4.71238951430128
x36 = -54.9778718424277
x37 = -98.9601690014881
x38 = 83.2522048029739
x39 = 14.1371671126075
x40 = 32.986723413003
x41 = -80.1106125773791
x42 = -17.2787595215844
x43 = -626.747734692733
x44 = -54.9778712582553
x45 = 45.5530935153116
x46 = -48.6946866728566
x47 = 89.5353905863307
x48 = 51.8362789048843
x49 = -42.4115006509881
x50 = 45.5530937462752
x51 = 26.7035373101781
x52 = 39.2699076443101
x53 = 76.9690196348227
x54 = -67.5442426460742
x54 = -67.5442426460742