How to Expand a Fraction Using Partial Fractions

In summary, the student attempted to solve a homework equation that used partial fractions, but got confused. They found the roots using the quadratic formula and simplified the terms, but are still confused.
  • #1
aerandir4
7
0

Homework Statement



expand by partial fractions:

Homework Equations



2(s+5)/(1.25*s^2+3s+9)

The Attempt at a Solution



ok I initially used the quadratic formula to get the two roots for the denominator
these being
(s+6/5+12i/5)(s+6/5-12i/5) i.e complex numbers

so now the partial fractions looks like this:

2(s+5)/(s+6/5+12i/5)(s+6/5-12i/5) = A/(s+6/5+12i/5)+B/(s+6/5-12i/5)

solving for B I get 1-19/12i which when multiplied by i/i = 1+19i/12 and A is the conjugate I believe, therefore A=1-19i/12

now the partial fraction looks like this
(1-19i/12)/(s+6/5-12i/5)+(1+19i/12)/(s+6/5+12i/5)

does this look right so far? If so how should I proceed in simplifying the terms?

thanks
 
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  • #2
Multiply the numerator and denominator of each fraction by the complex conjugate of its denominator.
 
  • #3
thanks,
have I worked it out right up to the last term?
 
  • #4
I haven't worked it out, but that suggestion might give you back what you started with.

Do you need to simplify it?
 
  • #5
Bill Foster said:
I haven't worked it out, but that suggestion might give you back what you started with.

Do you need to simplify it?

I personally prefer to always leave terms in the most simple of ways.
 
  • #6
aerandir4 said:
ok I initially used the quadratic formula to get the two roots for the denominator
these being
(s+6/5+12i/5)(s+6/5-12i/5) i.e complex numbers

so now the partial fractions looks like this:

2(s+5)/(s+6/5+12i/5)(s+6/5-12i/5) = A/(s+6/5+12i/5)+B/(s+6/5-12i/5)

Careful,

[tex]1.25*s^2+3s+9=\frac{5}{4}\left(s+\frac{6}{5}+\frac{12i}{5}\right)\left(s+\frac{6}{5}-\frac{12i}{5}\right)\neq\left(s+\frac{6}{5}+\frac{12i}{5}\right)\left(s+\frac{6}{5}-\frac{12i}{5}\right)[/itex]
 
  • #7
gabbagabbahey said:
Careful,

[tex]1.25*s^2+3s+9=\frac{5}{4}\left(s+\frac{6}{5}+\frac{12i}{5}\right)\left(s+\frac{6}{5}-\frac{12i}{5}\right)\neq\left(s+\frac{6}{5}+\frac{12i}{5}\right)\left(s+\frac{6}{5}-\frac{12i}{5}\right)[/itex]

I can't see how I've gone wrong. If you use the quadratic formula straight up with
a=1.25, b=3 and c=9 then

s1,s2=-3+-sqrt(9-45)/(5/2)

s1,s2=-3+-sqrt(-36)/(5/2)

s1,s2=-3+-6i/(5/2)
s1,s2=-6/5+-12i/5

so s+6/5-12i/5 and s+6/5+12i/5 are the two roots
 
  • #8
Sure, the roots are s1,s2=-3+-6i/(5/2), but as^2+bs+c=a(s-s1)(s-s2) not just (s-s1)(s-s2)
 
  • #9
gabbagabbahey said:
Sure, the roots are s1,s2=-3+-6i/(5/2), but as^2+bs+c=a(s-s1)(s-s2) not just (s-s1)(s-s2)

im getting a little confused now :confused:

I don't understand why its as^2+bs+c=a(s-s1)(s-s2).
Is this a special case or something? I have never seen it done like that before

at this stage... s1,s2=-6/5+-12i/5
don't you just take the whole term on the right to the left hand side?
where does the five over four come from here 5/4*(...)?
 
  • #10
It's basic algebra... when you expand (s-s1)(s-s2) using FOIL, you get s^2+bs/a+c/a not as^2+bs+c...you should really know this stuff by now
 

FAQ: How to Expand a Fraction Using Partial Fractions

What is the purpose of expanding by partial fractions?

The purpose of expanding by partial fractions is to break down a complex fraction into simpler fractions, making it easier to solve and manipulate in mathematical equations. This technique is commonly used in integration, differential equations, and algebraic manipulation.

How do you determine the partial fractions of a given fraction?

To determine the partial fractions of a given fraction, you must first factor the denominator into its linear or irreducible quadratic factors. Then, set up a system of equations by equating the original fraction to a sum of simpler fractions with undetermined coefficients. Finally, solve for the coefficients using various methods such as equating coefficients, substitution, or using a table of values.

What are the different types of partial fractions?

The two types of partial fractions are proper and improper fractions. Proper fractions have a smaller degree in the numerator than in the denominator, while improper fractions have a larger or equal degree in the numerator compared to the denominator. Improper fractions can be further classified into two types: repeated linear factors and irreducible quadratic factors.

Can all fractions be expanded by partial fractions?

No, not all fractions can be expanded by partial fractions. For a fraction to be expanded using this method, the denominator must be factorable into linear or irreducible quadratic factors. If the denominator has repeated linear factors or irreducible quadratic factors, then the fraction can be expanded by partial fractions.

What are the common mistakes to avoid when expanding by partial fractions?

Common mistakes to avoid when expanding by partial fractions include forgetting to factor the denominator, making errors in setting up the system of equations, and not checking for extraneous solutions or ensuring that all terms in the original fraction are accounted for in the partial fraction form. It is also important to ensure that the coefficients are correctly solved for and that the final expanded form is simplified as much as possible.

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