Question

How to solve and simplify

Answer 1

I am assuming that you meant "solve for x:  (x^2)/(x-4) + 16/(4-x)."  Unfortunately, this is an expression, not an equation.  Check that you have copied down this problem correctly.

I note right away that the 2nd term could be re-written as - 16/(x-4).  The advantage to this re-write is that the terms now have the same denominator.  You could combine these 2 terms as follows:  (x^2)/(x-4) - 16/(x-4), or

x^2 - 16
------------
   x-4
                                                                                     (x-4)(x+4)
The numerator factors easily; the correct result is      --------------
                                                                                           x-4

Simplifying, we get x+4.  Note:  this cannot be solved for x; it's just an expression.  This result is correct ONLY for x not equal to 4.  (Why?)


Now look at the 2nd problem.  Here you do have an equation involving fractions.  The LCD is (3)(4)=12.  Multiply all three terms of this equation by 12 to obtain

3(5x-2) -4(4x-5) = 12(1)=12.

Then 15x-6-16x+20=12, or -6+20-12=16x-15x = x.  Then x=2.  You MUST check this result via substituion before accepting it as the solution of this problem.