3/(x^2-9)-x/(x^2-6x+9)+1/(x+3)
3/((x+3)(x-3))- x/(x-3)(x-3) + 1/(x+3)
The LCD is (x-3)(x-3)(x+3)
= (3(x-3) - x(x+3) + 1/((x-3)(x-3))/((x-3)(x-3)(x+3))
= ....
take over
Hmm. I get
(3(x-3) - x(x+3) + 1(x-3)(x-3)) / ((x-3)(x-3)(x+3))
To simplify the expression 3/(x^2-9) - x/(x^2-6x+9) + 1/(x+3), we need to find a common denominator for all three fractions.
The denominators in the three fractions are (x^2-9), (x^2-6x+9), and (x+3). To find the common denominator, we need to factorize these expressions.
(x^2-9) = (x+3)(x-3)
(x^2-6x+9) = (x-3)(x-3) = (x-3)^2
Now, we can rewrite the expression using the common denominator:
[3/(x+3)(x-3)] - [x/(x-3)^2] + 1/(x+3)
Next, simplify each fraction:
3/(x+3)(x-3) - x/(x-3)^2 + 1/(x+3)
To subtract fractions, we need to have the same denominator. So, we need to rewrite the first fraction with the denominator of (x-3)^2.
To do this, we multiply the numerator and denominator of the first fraction by (x-3):
[3(x-3)/(x+3)(x-3)(x-3)] - x/(x-3)^2 + 1/(x+3)
Simplifying the first fraction:
[3x - 9/(x+3)(x-3)^2] - x/(x-3)^2 + 1/(x+3)
Now, we have a common denominator for all three fractions: (x-3)^2 * (x+3).
Combine the numerators:
[3x - 9 - x(x+3)] / [(x+3)(x-3)^2] + 1/(x+3)
Simplify and distribute:
(3x - 9 - x^2 - 3x) / [(x+3)(x-3)^2] + 1/(x+3)
Combine like terms:
(-x^2) / [(x+3)(x-3)^2] + 1/(x+3)
Now, let's simplify further:
-x^2 / [(x+3)(x-3)^2] + 1/(x+3)
To add or subtract fractions, we need to have the same denominator:
(-x^2 + (x-3)^2) / [(x+3)(x-3)^2]
And we can simplify the numerator:
(-x^2 + x^2 - 6x + 9) / [(x+3)(x-3)^2]
Simplify:
(-6x + 9) / [(x+3)(x-3)^2]
Therefore, the simplified expression is (-6x + 9) / [(x+3)(x-3)^2].