SOLUTION: deduce the solutions of the equation (z + 1)˄4 = 16(z - 1)˄4.

Algebra.Com
Question 1123522: deduce the solutions of the equation
(z + 1)˄4 = 16(z - 1)˄4.
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
(z + 1)^4 = 16(z - 1)^4
(z + 1)^4 = 2^4*(z - 1)^4
(z + 1)^4 = (2z - 2)^4
z+1 = 2z-2
z = 3
=================


Divide by (z-3)
---

By graphical methods (or by Newton's method) z = 1/3
------------
Divide by (z-1/3)


---
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

The discriminant -64 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -64 is + or - .

The solution is , or
Here's your graph:

z = (3/5) + 4i/5
z = (3/5) - 4i/5


Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!

Here's a better approach:


Use the principle of square roots:


Break that into two equations
; 
Use the principle of square roots again:
; 
; 

We solve the first using the + :
;




We solve the first using the - :
;




We solve the second using the + :
;











We could solve the second using the - the same way,
but since we know that if a polynomial equation with real
rational coefficients has a certain complex imaginary 
solution, then its conjugate is also a solution.

So  is also a solution.

The four solutions are:

, , , and 

Edwin
RELATED QUESTIONS

4(z-1)=16 (answered by edjones)
Please help me with the following. Thank you in advance. 1)state the domain and range of (answered by stanbon)
solve the equation z/z+1 = cos... (answered by ikleyn)
What is the greatest even number you can create by rearranging the digits of... (answered by Ed Parker)
solve the following equation 3/z-3 + 5/z-1 = 8/z-2 A) z=-4 B) z=-6 C) z=4 D)... (answered by edjones)
find all the solutions in the complex numbers, of z^4 + z^3 + z^2 + z + 1 = 0. use:... (answered by ikleyn)
Find all complex solutions to the equation z^8 + 16 = 17z^4 - 8z^6 - 8z^2. (answered by CPhill,ikleyn)
Perform the indicated operations and simplify. z-4/z-5-z+1/z+5+z-15/z^2-25 (answered by JBarnum)
Preform the indicated operations and then simplify z-4/z-5 - z+1/z+5 +... (answered by mananth)