$\displaystyle \frac{x+2}{2x-3}-\frac{x^2-4}{2x^2-3x}$
The problem is not simplifying correctly, somethings supposed to cross out and I'm not getting it
$\displaystyle \frac{x+2}{2x-3}-\frac{x^2-4}{2x^2-3x}$
Put under common denominator :
$\displaystyle \frac{(2x^2-3x)(x+2)}{(2x-3)(2x^2-3x)}-\frac{(2x-3)(x^2-4)}{(2x-3)(2x^2-3x)}$
Expand all this stuff :
$\displaystyle \frac{(2x^2-3x)(x+2) - (2x-3)(x^2-4)}{4x^3 - 6x^2 - 6x^2 + 9x}$
Expand further :
$\displaystyle \frac{(2x^3 + 4x^2 - 3x^2 - 6x) - (2x^3 - 8x - 3x^2 + 12)}{4x^3 - 12x^2 + 9x}$
Again :
$\displaystyle \frac{2x^3 + 4x^2 - 3x^2 - 6x - 2x^3 + 8x + 3x^2 - 12}{4x^3 - 12x^2 + 9x}$
Cancel out terms :
$\displaystyle \frac{4x^2 + 2x - 12}{4x^3 - 12x^2 + 9x}$
Factorize both polynomials :
$\displaystyle \frac{(4x - 6)(x + 2)}{x(4x - 6)(x - \frac{ 3}{2})}$
Cancel out :
$\displaystyle \frac{x + 2}{x(x - \frac{ 3}{2})}$
Finally :
$\displaystyle \frac{x + 2}{x^2 - \frac{ 3}{2}x}$
I think it's right, I haven't checked though ...
When adding/substracting fractions, all fractions need to be under the same denominator, otherwise it won't work. To achieve this, we usually multiply both parts of fraction 1 by the denominator of fraction 2, and the other way round. Then, we substract (in your case) the fractions. (we are not substracting the denominators, obviously)
EDIT : did you see that ? 77 posts, and 7 thanks ... it's my lucky day ...
No, we don't "usually multiply both parts of fraction 1 by the denominator of fraction 2, and the other way round." We usually multiply numerator and denominator of each fraction by whatever is necessary to ge the "least common denominator".
In the original fractions, the denominators are 2x-3 and $\displaystyle 2x^2- 3x= x(2x-3)$. The least common denominator is $\displaystyle x(2x-3)= 2x^2- 3x$. We need to multiply numerator and denominator of the first fraction, $\displaystyle \frac{x+2}{2x-3}$ by x to get $\displaystyle \frac{x^2+ 2x}{x(2x-3)}$. Since the denominator of the second fraction is already x(2x-3), we don't have to change that.
$\displaystyle \frac{x+2}{2x-3}- \frac{x^2- 4}{2x^2- 3x}= \frac{x^2+ 2x}{x(2x-3)}- \frac{x^2-4}{x(2x-3)}$$\displaystyle \frac{x^2+ 2x- x^2+ 4}{x(2x-3)}= \frac{2x+ 4}{x(2x-3)}$
You can, of course, multiply that denominator out to get $\displaystyle \frac{2x+4}{2x^2- 3x}$.
Ah yes, I didn't think of that.
Surely because when I was in junior I used to multiply the two denominators (typically 7 and 9) together to be sure to get a common divisor, because I wouldn't want to think about it.
I must think of doing this little check ... thanks HallsofIvy