Solving Partial Fractions: Finding Unknown Coefficients

In summary, the problem is finding the values of A and B in the equation (8x-17)/(x^2+x-12) = A/(x-3) + B/(x+4). The solution involves setting up and solving two equations with two unknowns, A and B. However, there may be differences in the values of A and B depending on how they are defined in the problem. The correct way to solve this problem is to plug in specific values for x and solve for A and B.
  • #1
PrudensOptimus
641
0
hi, the problem is:


∫ (8x-17)/x^2+x-12 dx = ∫ .../(x+4)(x-3) dx

so 8x - 17 = A(x-3) + B(x+4)

(A+B)x + 4B - 3A

so we have 2 eq and 2 unknown

A+B = 8
4B-3A = -17

... but the book says it's suppose to be 4A - 3B,.. I don't know what I did wrong.

Please help.
 
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  • #2
They probably just defined A and B differently than you did. (They must have said (8x-17)/(x2+x-12) = A/(x-3) + B/(x+4) ). You'll still get the right answer your way.
 
  • #3
the ending result is different...

they got B = 7 and A = 1...


I got B = 41/7, and A = 15/7
 
  • #4
Well, the B and A you listed aren't solutions to the equations you got...
 
  • #5
Originally posted by PrudensOptimus
(A+B)x + 4B - 3A

so we have 2 eq and 2 unknown

A+B = 8
4B-3A = -17

You don't have to solve these like that.

so 8x - 17 = A(x-3) + B(x+4)

Starting from here, just plug in x=+3 and x=-4.

x=3:
8(3)-17=A(3-3)=B(3+4)
7=7B
B=1

x=-4:
8(-4)-17=A(-4-3)+B(-4+4)
-49=-7A
A=7
 
  • #6
amazing.
 

1. What are partial fractions?

Partial fractions are mathematical expressions that involve a fraction with a polynomial in the numerator and denominator. They are used to simplify complex fractions and solve integrals.

2. How do you decompose a fraction into partial fractions?

To decompose a fraction into partial fractions, you need to factor the denominator into linear or irreducible quadratic factors. Then, for each unique factor, you set up a partial fraction with a numerator that is one degree less than the factor and solve for the coefficients.

3. What is the purpose of using partial fractions?

The main purpose of using partial fractions is to simplify complex fractions and solve integrals that involve fractions. By breaking down a fraction into smaller, simpler parts, it becomes easier to manipulate and solve.

4. Can all fractions be decomposed into partial fractions?

No, not all fractions can be decomposed into partial fractions. Only fractions with polynomials in the numerator and denominator can be broken down using partial fractions. Fractions with other types of functions, such as trigonometric functions, cannot be decomposed.

5. How are partial fractions used in real-world applications?

Partial fractions are used in a variety of real-world applications, such as in engineering, physics, and economics. They are particularly useful in solving differential equations and finding the inverse Laplace transform. In economics, they can be used to analyze supply and demand curves. In physics, they can be used to solve problems involving forces and motion.

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