f*(x)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the linear equation:
f*(x) = 1
Expand brackets in the left part
fx = 1
Divide both parts of the equation by f
x = 1 / (f)
We get the answer: x = 1/f
The solution of the parametric equation
Given the equation with a parameter:
$$f x = 1$$
Коэффициент при x равен
$$f$$
then possible cases for f :
$$f < 0$$
$$f = 0$$
Consider all cases in more detail:
With
$$f < 0$$
the equation
$$- x - 1 = 0$$
its solution
$$x = -1$$
With
$$f = 0$$
the equation
$$-1 = 0$$
its solution
no solutions
Sum and product of roots
[src]
re(f) I*im(f)
--------------- - ---------------
2 2 2 2
im (f) + re (f) im (f) + re (f)
$$\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
re(f) I*im(f)
--------------- - ---------------
2 2 2 2
im (f) + re (f) im (f) + re (f)
$$\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
re(f) I*im(f)
--------------- - ---------------
2 2 2 2
im (f) + re (f) im (f) + re (f)
$$\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
-I*im(f) + re(f)
----------------
2 2
im (f) + re (f)
$$\frac{\operatorname{re}{\left(f\right)} - i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
(-i*im(f) + re(f))/(im(f)^2 + re(f)^2)
re(f) I*im(f)
x1 = --------------- - ---------------
2 2 2 2
im (f) + re (f) im (f) + re (f)
$$x_{1} = \frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
x1 = re(f)/(re(f)^2 + im(f)^2) - i*im(f)/(re(f)^2 + im(f)^2)