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Equation:
Equation k^2+4=0
Equation 3x^2-28x+9=0
Equation 17-8y=4y-7
Equation ((8z-1)/5)-((3z+3)/4)=((50-2z)/9)+1
Express {x} in terms of y where:
15*x+15*y=16
-4*x+18*y=-14
8*x+7*y=-13
14*x-16*y=14
Identical expressions
2sin²пx+5sinпx- seven = zero
2 sinus of ²пx plus 5 sinus of пx minus 7 equally 0
2 sinus of ²пx plus 5 sinus of пx minus seven equally zero
2sin²пx+5sinпx-7=O
Similar expressions
2sin²пx-5sinпx-7=0
2sin²пx+5sinпx+7=0

2sin²пx+5sinпx-7=0 equation

The teacher will be very surprised to see your correct solution 😉

The solution

You have entered [src]

     2                            
2*sin (pi*x) + 5*sin(pi*x) - 7 = 0

\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0

Simplify

Detail solution

Given the equation

\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0

transform

2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)} - 7 = 0

\left(2 \sin^{2}{\left(\pi x \right)} + 5 \sin{\left(\pi x \right)}\right) - 7 = 0

Do replacement

w = \sin{\left(\pi 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 = 5

c = -7

, then

D = b^2 - 4 * a * c =

(5)^2 - 4 * (2) * (-7) = 81

Because D > 0, then the equation has two roots.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

w_{1} = 1

w_{2} = - \frac{7}{2}

do backward replacement

\sin{\left(\pi x \right)} = w

Given the equation

\sin{\left(\pi x \right)} = w

- this is the simplest trigonometric equation
This equation is transformed to

\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}

\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi

\pi x = 2 \pi n + \operatorname{asin}{\left(w \right)}

\pi x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi

, where n - is a integer
Divide both parts of the equation by

\pi

substitute w:

x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(w_{1} \right)}}{\pi}

x_{1} = \frac{2 \pi n + \operatorname{asin}{\left(1 \right)}}{\pi}

x_{1} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}

x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(w_{2} \right)}}{\pi}

x_{2} = \frac{2 \pi n + \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}

x_{2} = \frac{2 \pi n - \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}

x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi}{\pi}

x_{3} = \frac{2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi}{\pi}

x_{3} = \frac{2 \pi n + \frac{\pi}{2}}{\pi}

x_{4} = \frac{2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi}{\pi}

x_{4} = \frac{2 \pi n + \pi - \operatorname{asin}{\left(- \frac{7}{2} \right)}}{\pi}

x_{4} = \frac{2 \pi n + \pi + \operatorname{asin}{\left(\frac{7}{2} \right)}}{\pi}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

x1 = 1/2

x_{1} = \frac{1}{2}

     pi + re(asin(7/2))   I*im(asin(7/2))
x2 = ------------------ + ---------------
             pi                  pi

x_{2} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}

       re(asin(7/2))   I*im(asin(7/2))
x3 = - ------------- - ---------------
             pi               pi

x_{3} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}

x3 = -re(asin(7/2))/pi - i*im(asin(7/2))/pi

Sum and product of roots [src]

sum

1   pi + re(asin(7/2))   I*im(asin(7/2))     re(asin(7/2))   I*im(asin(7/2))
- + ------------------ + --------------- + - ------------- - ---------------
2           pi                  pi                 pi               pi

\left(\frac{1}{2} + \left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)

1   pi + re(asin(7/2))   re(asin(7/2))
- + ------------------ - -------------
2           pi                 pi

- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} + \frac{1}{2} + \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi}

product

pi + re(asin(7/2))   I*im(asin(7/2))                                    
------------------ + ---------------                                    
        pi                  pi       /  re(asin(7/2))   I*im(asin(7/2))\
------------------------------------*|- ------------- - ---------------|
                 2                   \        pi               pi      /

\frac{\frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}}{2} \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}}{\pi}\right)

-(I*im(asin(7/2)) + re(asin(7/2)))*(pi + I*im(asin(7/2)) + re(asin(7/2))) 
--------------------------------------------------------------------------
                                      2                                   
                                  2*pi

- \frac{\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{7}{2} \right)}\right)}\right)}{2 \pi^{2}}

-(i*im(asin(7/2)) + re(asin(7/2)))*(pi + i*im(asin(7/2)) + re(asin(7/2)))/(2*pi^2)

Numerical answer [src]

x1 = -37.5000001005768

x2 = -61.5000001005768

x3 = 86.4999998994232

x4 = -55.5000001005768

x5 = 6.49999989942322

x6 = -29.5000001005768

x7 = -83.5000001005768

x8 = 34.4999998994232

x9 = 40.4999998994232

x10 = -69.5000001005768

x11 = 52.4999998994232

x12 = -11.5000001005768

x13 = 46.4999998994232

x14 = -67.5000001005768

x15 = 88.4999998994232

x16 = 70.4999998994232

x17 = 28.4999998994232

x18 = 56.4999998994232

x19 = -49.5000001005768

x20 = -97.5000001005768

x21 = -25.5000001005768

x22 = 30.4999998994232

x23 = -35.5000001005768

x24 = -1.50000010057678

x25 = -21.5000001005768

x26 = -73.5000001005768

x27 = -79.5000001005768

x28 = 92.4999998994232

x29 = 24.4999998994232

x30 = -15.5000001005768

x31 = 16.4999998994232

x32 = 80.4999998994232

x33 = 84.4999998994232

x34 = 38.4999998994232

x35 = -63.5000001005768

x36 = 14.4999998994232

x37 = 60.4999998994232

x38 = -87.5000001005768

x39 = 68.4999998994232

x40 = -91.5000001005768

x41 = 42.4999998994232

x42 = -27.5000001005768

x43 = 36.4999998994232

x44 = -85.5000001005768

x45 = -65.5000001005768

x46 = -81.5000001005768

x47 = -53.5000001005768

x48 = -9.50000010057678

x49 = 4.49999989942322

x50 = -99.5000001005768

x51 = 66.4999998994232

x52 = 18.4999998994232

x53 = -59.5000001005768

x54 = 96.4999998994232

x55 = 0.499999899423223

x56 = 32.4999998994232

x57 = 78.4999998994232

x58 = -71.5000001005768

x59 = -5.50000010057678

x60 = 8.49999989942322

x61 = -19.5000001005768

x62 = -89.5000001005768

x63 = 48.4999998994232

x64 = -75.5000001005768

x65 = -13.5000001005768

x66 = -45.5000001005768

x67 = -77.5000001005768

x68 = 20.4999998994232

x69 = 10.4999998994232

x70 = -23.5000001005768

x71 = 74.4999998994232

x72 = 72.4999998994232

x73 = -57.5000001005768

x74 = 22.4999998994232

x75 = 94.4999998994232

x76 = -3.50000010057678

x77 = -51.5000001005768

x78 = -33.5000001005768

x79 = 64.4999998994232

x80 = 100.499999899423

x81 = -41.5000001005768

x82 = -93.5000001005768

x83 = -7.50000010057678

x84 = 26.4999998994232

x85 = 2.49999989942322

x86 = 50.4999998994232

x87 = 54.4999998994232

x88 = 12.4999998994232

x89 = -47.5000001005768

x90 = 76.4999998994232

x91 = 44.4999998994232

x92 = 58.4999998994232

x93 = -95.5000001005768

x94 = -17.5000001005768

x95 = 98.4999998994232

x96 = -43.5000001005768

x97 = 82.4999998994232

x98 = 62.4999998994232

x99 = 90.4999998994232

x100 = -31.5000001005768

x101 = -39.5000001005768

x101 = -39.5000001005768

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Identical expressions

Similar expressions

2sin²пx+5sinпx-7=0 equation

Numerical solution:

The solution