Question

Find the LCD of the given rational equation

Answer 1

`=(2x^3+x^2-11x+14)/(\left(x+4\right)\left(x+2\right)\left(x-2\right))`

Explanation:

Step 1: Factor
`x² + 2x - 8: (x - 2) (x + 4)`

`=(x)/(x+4)+(x-1)/(x+2)+(3)/(\left(x-2\right)\left(x+4\right))`

Step 2: Find the Least Common Multiplier of
`x + 4, x - 2, (x + 4), (x - 2)`

=
`(x + 4) (x + 2) (x - 2)`

Step 3a: Conform the fractions based on the LCM (Least Common Multiplier)

=
`(x\left(x+2\right)\left(x-2\right))/(\left(x+4\right)\left(x+2\right)\left(x-2\right))+(\left(x-1\right)\left(x+4\right)\left(x-2\right))/(\left(x+2\right)\left(x+4\right)\left(x-2\right))+(3\left(x+2\right))/(\left(x-2\right)\left(x+4\right)\left(x+2\right))`

Step 3b: The denominators are equal, so combine the fractions.

`=(x\left(x+2\right)\left(x-2\right)+\left(x-1\right)\left(x+4\right)\left(x-2\right)+3\left(x+2\right))/(\left(x+4\right)\left(x+2\right)\left(x-2\right))`

Step 4: Expand
`x(x + 2)(x - 2) +(x-1)(x+4)(x-2) + 3(x-2)`

`=(2x^3+x^2-11x+14)/(\left(x+4\right)\left(x+2\right)\left(x-2\right))`

Have a great day!