Simplify COmplex Fraction:
x-1
----
x-3
--------------------
1 4
------- - -----
x^2-x-6 x+2
Please Help!!
To simplify a complex fraction, we need to simplify both the numerator and denominator separately before dividing them. Let's break it down step by step.
First, let's simplify the numerator:
1. Factorize the numerator (x-1):
x - 1 = (x - 3) + 2
Now, let's simplify the denominator:
2. Factorize the denominator (x^2 - x - 6):
x^2 - x - 6 = (x - 3)(x + 2)
Now, write the expression in a simplified form using the factors we found above:
(x - 1)/(x - 3) = ((x - 3) + 2)/(x - 3) = (x - 3)/(x - 3) + 2/(x - 3) = 1 + 2/(x - 3)
1/(x^2 - x - 6) = 1/((x - 3)(x + 2))
Finally, divide the simplified numerator by the simplified denominator:
((x - 3) + 2)/(x - 3) ÷ 1/((x - 3)(x + 2)) = ((x - 3) + 2)/(x - 3) × ((x - 3)(x + 2))/1
= ((x - 3) + 2)(x - 3)(x + 2)/(x - 3)
= (x^2 - x - 6)(x + 2)/(x - 3)
= (x - 3)(x + 2)(x + 2)/(x - 3)
= (x + 2)^2
Thus, the simplified form of the given complex fraction is (x + 2)^2.