Help simplifying this equation

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In summary, the conversation is discussing the simplification and approximation of an original equation, u= [ (x^2 – 1)^1/2 + ln (x + (x^2 – 1)^1/2 )], by letting x=z+1. The first line that may cause confusion is (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5, which can be easily seen by factoring x^2 - 1 and simplifying. The second line, ln(1+e) approximately = e, can be derived through a Taylor series expansion around x = 0.
  • #1
ComFlu945
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I'm trying to follow some notes in class.

so the original equation is:
u= [ (x^2 – 1)^1/2 + ln (x + (x^2 – 1)^1/2 )]

and we let x=z+1.

Then by simplifying and approximating, u=z*2*2^.5 + higher order terms

from my notes: (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5
ln(1+e) approximately = e
 
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  • #2
ComFlu945 said:
from my notes: (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5
ln(1+e) approximately = e

I presume those are the two lines that have you confused?

The first one is most easily seen by noting that
x2 - 1 = (x + 1)(x - 1)
which, if z = x - 1, is z(z + 2). Then the algebra is straightforward (take out a factor of 2 and split the root of product into product of roots).


For the second one, you can make a Taylor series expansion around x = 0 of
ln(1 + x) = ln(1) + 1/(1 + 0) x + O(x2)
and see that to first order, you get x.
 

Related to Help simplifying this equation

Question 1: What is the purpose of simplifying an equation?

Simplifying an equation involves reducing it to its most basic and concise form. This makes it easier to understand and work with, and can help in solving problems or making calculations.

Question 2: How do I know when an equation needs to be simplified?

An equation may need to be simplified if it contains unnecessary or redundant terms, or if it is too complex to easily work with. In general, it is good practice to simplify equations whenever possible.

Question 3: What are the steps involved in simplifying an equation?

The specific steps for simplifying an equation may vary depending on the complexity and type of equation. However, some common techniques include combining like terms, factoring, and using properties of exponents.

Question 4: Can simplifying an equation change its solution?

Simplifying an equation involves algebraic manipulations, but it should not change the overall solution. However, it is always important to check the validity of the solution after simplifying.

Question 5: Are there any resources available to help with simplifying equations?

Yes, there are many online resources and textbooks that provide step-by-step instructions and practice problems for simplifying equations. It can also be helpful to seek guidance from a math tutor or teacher if needed.

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