Question
Solve the system of equations
z∈/R
Alternative Form
No real solution
Evaluate
{z×1=23iz2=1−2i
Any expression multiplied by 1 remains the same
{z=23iz2=1−2i
Calculate
More Steps

Evaluate
3iz2=1−2i
Simplify
z=1−2i
Rewrite the complex number in polar form
More Steps

Evaluate
1−2i
Determine the modulus and the argument of the complex number
r=12+(−2)2θ=arctan(1−2)
Calculate
r=5θ=arctan(1−2)
Since 1−2i lies in the IV quadrant, add 2π to get the argument in the IV quadrant
r=5θ=arctan(−2)+2π
Substitute the given values into the formula r(cosθ+isinθ)
5×(cos(arctan(−2)+2π)+isin(arctan(−2)+2π))
z=5×(cos(arctan(−2)+2π)+isin(arctan(−2)+2π))
Calculate the nth roots of a complex r(cos(θ)+i×sin(θ),using nz=nr(cosnθ+2kπ+isinnθ+2kπ)
z=5×(cos(2arctan(−2)+2π+2kπ)+isin(2arctan(−2)+2π+2kπ))
Simplify
z=45×(cos(2arctan(−2)+2π+2kπ)+isin(2arctan(−2)+2π+2kπ))
Since n=2,substitute k=0,1 into the expression
z1=45×(cos(2arctan(−2)+2π+2×0×π)+isin(2arctan(−2)+2π+2×0×π))z2=45×(cos(2arctan(−2)+2π+2×1×π)+isin(2arctan(−2)+2π+2×1×π))
Calculate
More Steps

Evaluate
2arctan(−2)+2π+2×0×π
Any expression multiplied by 0 equals 0
2arctan(−2)+2π+0
Removing 0 doesn't change the value,so remove it from the expression
2arctan(−2)+2π
z1=45×(cos(2arctan(−2)+2π)+isin(2arctan(−2)+2π))z2=45×(cos(2arctan(−2)+2π+2×1×π)+isin(2arctan(−2)+2π+2×1×π))
Calculate
z1=45×(cos(2arctan(−2)+2π)+isin(2arctan(−2)+2π))z2=45×(cos(2arctan(−2)+2π+2π)+isin(2arctan(−2)+2π+2π))
Calculate
z1=−45×cos(−2arctan(−2))+45×sin(2arctan(−2)+2π)×i∪z2=45×cos(2arctan(−2)+2π+2π)+45×sin(2arctan(−2)+2π+2π)×i
{z=2z1=−45×cos(−2arctan(−2))+45×sin(2arctan(−2)+2π)×i∪z2=45×cos(2arctan(−2)+2π+2π)+45×sin(2arctan(−2)+2π+2π)×i
Solution
z∈/R
Alternative Form
No real solution
Show Solution
