Adding and subtracting rational expressions.
3x/x^2+x-12 - x/x^2-16
Online you needed parentheses so things are not ambiguous. You want a common denominator, so note that
x^2+x-12 = (x+4)(x-3)
x^2-16 = (x+4)(x-4)
Now you can see that the LCD is
(x+4)(x-4)(x-3)
and work with that.
i would have never figured this out
To add or subtract rational expressions, you need to find a common denominator. Let's look at the given problem step by step:
Expression 1: 3x / (x^2 + x - 12)
Expression 2: -x / (x^2 - 16)
First, let's factor the denominators:
Expression 1 denominator: (x^2 + x - 12) = (x + 4)(x - 3)
Expression 2 denominator: (x^2 - 16) = (x - 4)(x + 4)
Now we can determine the least common denominator (LCD), which is the product of the factors in both denominators:
LCD = (x + 4)(x - 3)(x - 4)
To write each expression with the common denominator, we need to multiply the numerator and denominator of each expression by any missing factors:
Expression 1: (3x / (x^2 + x - 12)) * ((x - 4) / (x - 4)) = (3x(x - 4)) / ((x + 4)(x - 3)(x - 4))
Expression 2: (-x / (x^2 - 16)) * ((x - 3) / (x - 3)) = (-x(x - 3)) / ((x - 4)(x + 4)(x - 3))
Now that both expressions have the same denominator, we can combine them:
(3x(x - 4) - x(x - 3)) / ((x + 4)(x - 3)(x - 4))
Simplifying the numerators:
(3x^2 - 12x - x^2 + 3x) / ((x + 4)(x - 3)(x - 4))
Combining like terms in the numerator:
(2x^2 - 9x) / ((x + 4)(x - 3)(x - 4))
This is the simplified form of the sum (or difference) of the given rational expressions.