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- Equation:
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Equation 2*x^2+6*x+9=0
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Equation 10(x-9)=7
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Equation (x+2)^4+(x+2)^2-12=0
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Equation (x^2-4)^2+(x^2-3*x-10)^2=0
- Express {x} in terms of y where:
- 4*x-18*y=-14
- -14*x-1*y=3
- -3*x-6*y=18
- 10*x+16*y=-6
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Identical expressions
- 2sin^2x+7cosx- five = zero
- 2 sinus of squared x plus 7 co sinus of e of x minus 5 equally 0
- 2 sinus of squared x plus 7 co sinus of e of x minus five equally zero
- 2sin2x+7cosx-5=0
- 2sin²x+7cosx-5=0
- 2sin to the power of 2x+7cosx-5=0
- 2sin^2x+7cosx-5=O
-
Similar expressions
- 2sin^2x+7cosx+5=0
- 2sin^2x-7cosx-5=0
2sin^2x+7cosx-5=0 equation
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The solution
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[src]
2 2*sin (x) + 7*cos(x) - 5 = 0
$$\left(2 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)}\right) - 5 = 0$$
Detail solution
Given the equation
$$\left(2 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)}\right) - 5 = 0$$
transform
$$7 \cos{\left(x \right)} - \cos{\left(2 x \right)} - 4 = 0$$
$$- 2 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 3 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = -2$$
$$b = 7$$
$$c = -3$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 3$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{2 \pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
$$\left(2 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)}\right) - 5 = 0$$
transform
$$7 \cos{\left(x \right)} - \cos{\left(2 x \right)} - 4 = 0$$
$$- 2 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 3 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = -2$$
$$b = 7$$
$$c = -3$$
, then
D = b^2 - 4 * a * c =
(7)^2 - 4 * (-2) * (-3) = 25
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{2}$$
$$w_{2} = 3$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(3 \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{3} = \pi n - \frac{2 \pi}{3}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(3 \right)}$$
Rapid solution
[src]
-pi
x1 = ----
3
$$x_{1} = - \frac{\pi}{3}$$
pi
x2 = --
3
$$x_{2} = \frac{\pi}{3}$$
/ ___\
|\/ 2 |
x3 = -2*I*atanh|-----|
\ 2 /
$$x_{3} = - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
/ ___\
|\/ 2 |
x4 = 2*I*atanh|-----|
\ 2 /
$$x_{4} = 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
x4 = 2*i*atanh(sqrt(2)/2)
Sum and product of roots
[src]
sum
/ ___\ / ___\ pi pi |\/ 2 | |\/ 2 | - -- + -- - 2*I*atanh|-----| + 2*I*atanh|-----| 3 3 \ 2 / \ 2 /
$$\left(\left(- \frac{\pi}{3} + \frac{\pi}{3}\right) - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}\right) + 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
0
$$0$$
product
/ ___\ / ___\ -pi pi |\/ 2 | |\/ 2 | ----*--*-2*I*atanh|-----|*2*I*atanh|-----| 3 3 \ 2 / \ 2 /
$$2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)} - 2 i \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)} - \frac{\pi}{3} \frac{\pi}{3}$$
=
/ ___\
2 2|\/ 2 |
-4*pi *atanh |-----|
\ 2 /
--------------------
9
$$- \frac{4 \pi^{2} \operatorname{atanh}^{2}{\left(\frac{\sqrt{2}}{2} \right)}}{9}$$
-4*pi^2*atanh(sqrt(2)/2)^2/9
Numerical answer
[src]
x1 = 36.6519142918809
x2 = -24.0855436775217
x3 = 11.5191730631626
x4 = -32.4631240870945
x5 = 32.4631240870945
x6 = 26.1799387799149
x7 = 49.2182849062401
x8 = -7.33038285837618
x9 = 99.4837673636768
x10 = 17.8023583703422
x11 = -95.2949771588904
x12 = -42.9350995990605
x13 = -893.259511170698
x14 = -68.0678408277789
x15 = 19.8967534727354
x16 = 45.0294947014537
x17 = -26.1799387799149
x18 = -86.9173967493176
x19 = 80.634211442138
x20 = 57.5958653158129
x21 = 7.33038285837618
x22 = 24.0855436775217
x23 = 95.2949771588904
x24 = -19.8967534727354
x25 = -13.6135681655558
x26 = -30.3687289847013
x27 = 38.7463093942741
x28 = 82.7286065445312
x29 = 5.23598775598299
x30 = -45.0294947014537
x31 = 1.0471975511966
x32 = -1.0471975511966
x33 = -93.2005820564972
x34 = -5.23598775598299
x35 = -11.5191730631626
x36 = -74.3510261349584
x37 = -57.5958653158129
x38 = 42.9350995990605
x39 = 86.9173967493176
x40 = 76.4454212373516
x41 = -80.634211442138
x42 = 55.5014702134197
x43 = -89.0117918517108
x44 = -38.7463093942741
x45 = -17.8023583703422
x46 = 89.0117918517108
x47 = -51.3126800086333
x48 = -55.5014702134197
x49 = 63.8790506229925
x50 = 68.0678408277789
x51 = -76.4454212373516
x52 = 51.3126800086333
x53 = -99.4837673636768
x54 = -49.2182849062401
x55 = -36.6519142918809
x56 = -63.8790506229925
x57 = 61.7846555205993
x58 = 13.6135681655558
x59 = 799.011731563004
x60 = 74.3510261349584
x61 = -70.162235930172
x62 = 93.2005820564972
x63 = 70.162235930172
x64 = -82.7286065445312
x65 = -61.7846555205993
x66 = 30.3687289847013
x66 = 30.3687289847013