Mister Exam
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How to use it?
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Similar expressions
- 14/x-3,14x=0
14/x+3,14x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\frac{157 x}{50} + \frac{14}{x} = 0$$
transform
$$x^{2} = - \frac{700}{157}$$
Because equation degree is equal to = 2 and the free term = -700/157 < 0,
so the real solutions of the equation d'not exist
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{2} = - \frac{700}{157}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{2} e^{2 i p} = - \frac{700}{157}$$
where
$$r = \frac{10 \sqrt{1099}}{157}$$
- the magnitude of the complex number
Substitute r:
$$e^{2 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(2 p \right)} + \cos{\left(2 p \right)} = -1$$
so
$$\cos{\left(2 p \right)} = -1$$
and
$$\sin{\left(2 p \right)} = 0$$
then
$$p = \pi N + \frac{\pi}{2}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{10 \sqrt{1099} i}{157}$$
$$z_{2} = \frac{10 \sqrt{1099} i}{157}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{10 \sqrt{1099} i}{157}$$
$$x_{2} = \frac{10 \sqrt{1099} i}{157}$$
$$\frac{157 x}{50} + \frac{14}{x} = 0$$
transform
$$x^{2} = - \frac{700}{157}$$
Because equation degree is equal to = 2 and the free term = -700/157 < 0,
so the real solutions of the equation d'not exist
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{2} = - \frac{700}{157}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{2} e^{2 i p} = - \frac{700}{157}$$
where
$$r = \frac{10 \sqrt{1099}}{157}$$
- the magnitude of the complex number
Substitute r:
$$e^{2 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(2 p \right)} + \cos{\left(2 p \right)} = -1$$
so
$$\cos{\left(2 p \right)} = -1$$
and
$$\sin{\left(2 p \right)} = 0$$
then
$$p = \pi N + \frac{\pi}{2}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{10 \sqrt{1099} i}{157}$$
$$z_{2} = \frac{10 \sqrt{1099} i}{157}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{10 \sqrt{1099} i}{157}$$
$$x_{2} = \frac{10 \sqrt{1099} i}{157}$$
Sum and product of roots
[src]
sum
______ ______
10*I*\/ 1099 10*I*\/ 1099
- ------------- + -------------
157 157
$$- \frac{10 \sqrt{1099} i}{157} + \frac{10 \sqrt{1099} i}{157}$$
=
0
$$0$$
product
______ ______
-10*I*\/ 1099 10*I*\/ 1099
--------------*-------------
157 157
$$- \frac{10 \sqrt{1099} i}{157} \frac{10 \sqrt{1099} i}{157}$$
=
700 --- 157
$$\frac{700}{157}$$
700/157
Rapid solution
[src]
______
-10*I*\/ 1099
x1 = --------------
157
$$x_{1} = - \frac{10 \sqrt{1099} i}{157}$$
______
10*I*\/ 1099
x2 = -------------
157
$$x_{2} = \frac{10 \sqrt{1099} i}{157}$$
x2 = 10*sqrt(1099)*i/157