Problem 3:
4x/(x+2) - 8/(x-1)
What I did:
Common denominator is: (x+2)(x-1)
4x(x-1)/(x+2)(x-1) - 8(x+2)/(x+2)(x-1)
[4x(x-1) - 8(x+2)]/(x+2)(x-1)
multiply the numerator out.
4x^2 -4x -8x-16
4x^2-12x-16 factor 4 out
4(x^2-3x-4)
factor
4(x-4)(x+1)
and that will reduce over the denominator.
To solve Problem 3, you need to simplify the given expression step by step. Here's a breakdown of the solution process:
Given expression: 4x/(x+2) - 8/(x-1)
1. Find the common denominator:
The common denominator is the product of the two denominators, (x+2)(x-1).
2. Convert each fraction with the common denominator:
Multiply the numerator and the denominator of the first fraction, 4x/(x+2), by (x-1). Multiply the numerator and the denominator of the second fraction, 8/(x-1), by (x+2).
3. Combine the fractions with the common denominator:
Now that both fractions have the same denominator, you can subtract them. Rewrite the expression as [4x(x-1) - 8(x+2)]/(x+2)(x-1).
4. Multiply the numerator out:
Simplify the numerator by multiplying out the expressions within the brackets. You have 4x(x-1) - 8(x+2), which simplifies to 4x^2 - 4x - 8x - 16.
5. Arrange the terms:
Rearrange the terms in descending order, so you have 4x^2 - 12x -16.
6. Factor out the common factor:
Factor out a 4 from the expression, giving you 4(x^2 - 3x - 4).
7. Factor the quadratic expression:
The quadratic expression x^2 - 3x - 4 is factorable as (x-4)(x+1).
8. Simplify the expression:
Now simplify the factored expression by substituting the factors back into the expression. You get 4(x-4)(x+1).
This is the simplified form of the given expression, and it cannot be further reduced.