SOLUTION: 1) if x is a rational number and (x+1)^3 -(x-1)^3 / (x+1)^2 - (x-1)^2 = 2, then the sum of numerator and denominator of x is ? 2) if x^2 + y^2 + 1 = 2x, then x^3 + y^5 = ?

Question 973934: 1) if x is a rational number and (x+1)^3 -(x-1)^3 / (x+1)^2 - (x-1)^2 = 2, then the sum of numerator and denominator of x is ?
2) if x^2 + y^2 + 1 = 2x, then x^3 + y^5 = ?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
2) If x^2+y^2+1=2x, then
x^2+y^2+1-2x=0 ---> (x^2-2x+1)+y^2=0 ---> (x-1)^2+y^2=0 ---> y=0 and x-1=0<-->x=1.
Then,
x^3 + y^5 = 1^2 + 0^2 =1 + 0 = 1

1) Did you really mean
(x+1)^3 -(x-1)^3 / (x+1)^2 - (x-1)^2 == 2?
Or did you mean
[ (x+1)^3 -(x-1)^3 ] / [ (x+1)^2 - (x-1)^2 ]== 2?
I like the expression,
because it has a sort of elegant symmetry, and simplifies nicely.

You can expand all those cubes and squares and laboriously simplify.
We can also use special products to simplify the expression with less risk for mistakes.
If we make the to save some writing, we can simplify easier.
We know that , that , and that .
So, .
So, going back to ,
==
=

, so if = 2,
-->-->-->--> .
Then, the sum of numerator and denominator of x is either or .

=
=
So, if , then
-->
-->-->
-->-->

The rational number ,
where and are integers with no common factors,
that could be a solution to the equation above
has an that is a factor of the independent term, ,
and an which is a factor of the leading coefficient, .
So, or .
The problem with that, is that
for , ,
and makes the denominator zero.
For , , and
is undefined.
My conclusion is that there is no rational value of that will make
= 2
RELATED QUESTIONS
The denominator of a fraction in its simplest form is greater than the numerator by 5. If (answered by greenestamps)

Solve the inequality and graph the solution set on a real number line. (1/x+1) >... (answered by Earlsdon)

if x+1/x= 2 then what is the value of... (answered by tommyt3rd)

if x=sqrt{3+2sqrt(2)} then find the value of x^2+1/x^2 and... (answered by mananth)

This is actually a Calculus 1 problem, but I am having problems with the algebra portion. (answered by solver91311)

what is the product of (x-y)(x^2-xy+y^2)? and then what is the product of... (answered by unlockmath)

A rational number becomes 2 / 3 when 1 is added to its numerator and the denominator. But (answered by josgarithmetic)

solve 6/(x^2-9)- 1/(x-3) = 1 the 6 is the numerator and (x^2-9) is the... (answered by jim_thompson5910)

Simply and express your answer with a denominator of 1 : {{{... (answered by ikleyn)