Mister Exam
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How to use it?
- Equation:
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Equation x^2-4x+13=0
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Equation x^3+x^2-9*x-9=0
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Equation 3*x-5=x+7
- Equation x^3+12*x^2-4*x-48=0
- Express {x} in terms of y where:
- -11*x+3*y=5
- 10*x-18*y=-6
- -2*x+17*y=-3
- -2*x+13*y=20
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Identical expressions
- Iog(x^ two -x)+ six =t
- Iog(x squared minus x) plus 6 equally t
- Iog(x to the power of two minus x) plus six equally t
- Iog(x2-x)+6=t
- Iogx2-x+6=t
- Iog(x²-x)+6=t
- Iog(x to the power of 2-x)+6=t
- Iogx^2-x+6=t
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Similar expressions
- Iog(x^2+x)+6=t
- Iog(x^2-x)-6=t
Iog(x^2-x)+6=t equation
The teacher will be very surprised to see your correct solution 😉
The solution
The graph
Rapid solution
[src]
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/ 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\
4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /|
\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------|
1 \ 2 / \ 2 /
x1 = - - ---------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------
2 2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}$$
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/ 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\
4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /|
\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------|
1 \ 2 / \ 2 /
x2 = - + ---------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------
2 2 2
$$x_{2} = \frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}$$
x2 = i*((4*exp(re(t) - 6)*cos(im(t)) + 1)^2 + 16*exp(2*re(t) - 12)*sin(im(t))^2)^(1/4)*sin(atan2(4*exp(re(t) - 6)*sin(im(t), 4*exp(re(t) - 6)*cos(im(t)) + 1)/2)/2 + ((4*exp(re(t) - 6)*cos(im(t)) + 1)^2 + 16*exp(2*re(t) - 12)*sin(im(t))^2)^(1/4)*cos(atan2(4*exp(re(t) - 6)*sin(im(t)), 4*exp(re(t) - 6)*cos(im(t)) + 1)/2)/2 + 1/2)
Sum and product of roots
[src]
sum
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/ 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\
4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /|
\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------| \/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------|
1 \ 2 / \ 2 / 1 \ 2 / \ 2 /
- - ---------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + - + ---------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}\right) + \left(\frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}\right)$$
=
1
$$1$$
product
/ _________________________________________________________________ _________________________________________________________________ \ / _________________________________________________________________ _________________________________________________________________ \ | / 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\| | / 2 / / -6 + re(t) -6 + re(t)\\ / 2 / / -6 + re(t) -6 + re(t)\\| | 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /|| | 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /| 4 / / -6 + re(t)\ 2 -12 + 2*re(t) |atan2\4*e *sin(im(t)), 1 + 4*cos(im(t))*e /|| | \/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------|| | \/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *cos|-------------------------------------------------------------| I*\/ \1 + 4*cos(im(t))*e / + 16*sin (im(t))*e *sin|-------------------------------------------------------------|| |1 \ 2 / \ 2 /| |1 \ 2 / \ 2 /| |- - ---------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------|*|- + ---------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------| \2 2 2 / \2 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}\right) \left(\frac{i \sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} + 16 e^{2 \operatorname{re}{\left(t\right)} - 12} \sin^{2}{\left(\operatorname{im}{\left(t\right)} \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)},4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1 \right)}}{2} \right)}}{2} + \frac{1}{2}\right)$$
=
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/ 2*re(t) 6 + re(t) 12 -6 / 2*re(t) 6 + re(t) 12 -12 + re(t) / 2*re(t) 6 + re(t) 12 -6 + re(t)
1 \/ 16*e + 8*cos(im(t))*e + e *e \/ 16*e + 8*cos(im(t))*e + e *cos(im(t))*e I*\/ 16*e + 8*cos(im(t))*e + e *e *sin(im(t))
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4 ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________
/ 2 / 2 / 2
/ / -6 + re(t)\ -12 + 2*re(t) -12 + 2*re(t) / / -6 + re(t)\ -12 + 2*re(t) -12 + 2*re(t) / 2*re(t) / -6 + re(t)\ 12 2*re(t)
4*\/ \1 + 4*cos(im(t))*e / + 8*e - 8*cos(2*im(t))*e \/ \1 + 4*cos(im(t))*e / + 8*e - 8*cos(2*im(t))*e \/ 8*e + \1 + 4*cos(im(t))*e / *e - 8*cos(2*im(t))*e
$$- \frac{\sqrt{8 e^{\operatorname{re}{\left(t\right)} + 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 16 e^{2 \operatorname{re}{\left(t\right)}} + e^{12}} e^{\operatorname{re}{\left(t\right)} - 12} \cos{\left(\operatorname{im}{\left(t\right)} \right)}}{\sqrt{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} - 8 e^{2 \operatorname{re}{\left(t\right)} - 12} \cos{\left(2 \operatorname{im}{\left(t\right)} \right)} + 8 e^{2 \operatorname{re}{\left(t\right)} - 12}}} - \frac{\sqrt{8 e^{\operatorname{re}{\left(t\right)} + 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 16 e^{2 \operatorname{re}{\left(t\right)}} + e^{12}}}{4 \sqrt{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} - 8 e^{2 \operatorname{re}{\left(t\right)} - 12} \cos{\left(2 \operatorname{im}{\left(t\right)} \right)} + 8 e^{2 \operatorname{re}{\left(t\right)} - 12}} e^{6}} + \frac{1}{4} - \frac{i \sqrt{8 e^{\operatorname{re}{\left(t\right)} + 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 16 e^{2 \operatorname{re}{\left(t\right)}} + e^{12}} e^{\operatorname{re}{\left(t\right)} - 6} \sin{\left(\operatorname{im}{\left(t\right)} \right)}}{\sqrt{\left(4 e^{\operatorname{re}{\left(t\right)} - 6} \cos{\left(\operatorname{im}{\left(t\right)} \right)} + 1\right)^{2} e^{12} - 8 e^{2 \operatorname{re}{\left(t\right)}} \cos{\left(2 \operatorname{im}{\left(t\right)} \right)} + 8 e^{2 \operatorname{re}{\left(t\right)}}}}$$
1/4 - sqrt(16*exp(2*re(t)) + 8*cos(im(t))*exp(6 + re(t)) + exp(12))*exp(-6)/(4*sqrt((1 + 4*cos(im(t))*exp(-6 + re(t)))^2 + 8*exp(-12 + 2*re(t)) - 8*cos(2*im(t))*exp(-12 + 2*re(t)))) - sqrt(16*exp(2*re(t)) + 8*cos(im(t))*exp(6 + re(t)) + exp(12))*cos(im(t))*exp(-12 + re(t))/sqrt((1 + 4*cos(im(t))*exp(-6 + re(t)))^2 + 8*exp(-12 + 2*re(t)) - 8*cos(2*im(t))*exp(-12 + 2*re(t))) - i*sqrt(16*exp(2*re(t)) + 8*cos(im(t))*exp(6 + re(t)) + exp(12))*exp(-6 + re(t))*sin(im(t))/sqrt(8*exp(2*re(t)) + (1 + 4*cos(im(t))*exp(-6 + re(t)))^2*exp(12) - 8*cos(2*im(t))*exp(2*re(t)))