What is the least common factor in equation $\frac{5}{x+4}=4+\frac{3}{x-2}$ I am attempting to solve for x $\frac{5}{x+4}=4+\frac{3}{x-2}$
I know that I need to find the least common denominator. In this case, since I cannot see a clear relationship among them all I think it's just the product of all 3 denominators:
$\frac{5}{x+4}=4+\frac{3}{x-2}$ = $\frac{5}{x+4}=\frac{4}{1}+\frac{3}{x-2}$
LCD: $(x+4)(1)(x-2)$ = $(x+4)(x-2)$
Is this the LCD?
Because I tried to use this in solving my equation but I arrived at a quadratic. I don't think that my textbook wants me to use quadratics in this section but I'm not sure. Here's how I arrived at that:
$\frac{5}{x+4}=\frac{4}{1}+\frac{3}{x-2}$
$(x+4)(x-2)\frac{5}{x+4}=(x+4)(x-2)\frac{4}{1}+(x+4)(x-2)\frac{3}{x-2}$
Then cancel out common factors:
$(x-2)5=(x+4)(x-2)(4)+(x+4)(3)$
$5x-10=(x+4)(x-2)(4)+3x+12$
And if I multiple out the middle term I'll get a polynomial, which is unexpected so I'm not sure I'm on the right path here... am I?
$5x-10=4x^2-4x-8+3x+12$
$5x-10=4x^2-x+4$

Yes, you are on the right path. Yes, you do get a quadratic. The quadratic that you should get is $10-6x-4x^2$ and its roots are $1$ and $-\frac52$.
