Algebra stuff

[kengr]

Below is my attempt to solve a sample algebra problem for griffen

I'd like folks to look it over to see if I did it right, and also have them try to explain *why* I did what I did at each step. Alternate processes for solving are welcome too.

-4                       3
------------  = - ---- -2
2x^2-x+3         x-1

There are several approaches possible. We can combine the terms on the right. We can multiply both sides by the bottom of the fraction on the left. and there are probably other approaches as well.

I'm going to start by combining the terms on the right

    3
- ---- -2
  x-1

The lowest common denominator is (x-1)*1

         x-1      2(x-1)
2 * -----  =  ---------
         x-1         x-1

Which makes the right side

     3           2(x-1)
- ------  -  ---------
   x-1           x-1

that becomes

     3 - 2(x-1)
- --------------
        x-1 

And the overall equation is now

-4                       3 - 2(x-1)
------------  = - --------------
2x^2-x+3              x-1

Now we multiply both sides by x-1

-4(x-1)              3(x-1) - 2(x-1)^2
------------  = - --------------
2x^2-x+3              x-1

Simplify the right side:

-4(x-1)
------------  = -  ( 3 - 2(x-1)) 
2x^2-x+3    

Multiply both sides by -1 (to get rid of those annoying - signs)

4(x-1)
------------  =  3 - 2(x-1)
2x^2-x+3    

Now I can either multiply both sides by (2x^2-x+3) or I can divide both the top and bottom of the left side by (x-1)

I've forgotten how to do that sort of division, and I'm not sure I can factor the (2x^2-x+3). So I'll go with the multiplication.

4(x-1) = (3 - 2(x-1))(2x^2-x+3)

ok, let's simplify the sides a bit.

4x-4 = (3-2x-2)(2x^2-x+3)

4x-4 = (2x+1)(2x^2-x+3)

Now to multiply out the right side. We have two terms times 3 terms, so before simplification, we'll have 6 terms in the answer.

4x-4 = (2x*2x^2) + (2x*-x) + (2x*3) + (1*2x^2) + (1*-x) + (1*3)

Do the stuff in the parentheses.

4x-4 = 4x^3         + -2x^2    + 6x        +  2x^2       +  -x      + 3

Simplify the signs

4x-4 = 4x^3         -2x^2       + 6x        + 2x^2        -x           +3

Group by order (ie x^3 terms together, x^2 terms together, etc)

4x-4 = 4x^3         -2x^2  + 2x^2     + 6x -x           +3

Combine same order terms

4x-4 = 4x^3         0     + 5x    +3

Finish simplification

4x-4 = 4x^3 + 5x +3

subtract 4x from both sides

 -4 = 4x^3+x+3

Add 4 to both sides

0 = 4x^3 +x +7