Weird Rational Equation

[nycmathguy]

Solve for x.

(x^3 + x)/[(x^2 - x + 1)^2] = 10/9

 

[MathLover1]

(x^3 + x)/(x^2 - x + 1)^2= 10/9...........cross multiply

9(x^3 + x)= 10(x^2 - x + 1)^2
9x^3 + 9x= 10(x^4 - 2x^3 + 3 x^2 - 2 x + 1)
9x^3 + 9x= 10x^4 - 20x^3 + 30x^2 - 20x + 10
10x^4 - 20x^3 -9x^3+ 30x^2 - 20x- 9x+ 10=0
10x^4 - 29x^3 + 30x^2 - 29x + 10=0..........factor
(2 x - 1) (x - 2) (5 x^2 - 2 x + 5)=0

solutions:

if (2 x - 1) =0=> 2x=1=>x=1/2
if (x - 2)=0=>x=2
if (5 x^2 - 2 x + 5)=0 =>x = 1/5 - (2i*sqrt(6))/5 or x = 1/5 + (2i*sqrt(6))/5

 

[nycmathguy]

(x^3 + x)/(x^2 - x + 1)^2= 10/9...........cross multiply

9(x^3 + x)= 10(x^2 - x + 1)^2
9x^3 + 9x= 10(x^4 - 2x^3 + 3 x^2 - 2 x + 1)
9x^3 + 9x= 10x^4 - 20x^3 + 30x^2 - 20x + 10
10x^4 - 20x^3 -9x^3+ 30x^2 - 20x- 9x+ 10=0
10x^4 - 29x^3 + 30x^2 - 29x + 10=0..........factor
(2 x - 1) (x - 2) (5 x^2 - 2 x + 5)=0

solutions:

if (2 x - 1) =0=> 2x=1=>x=1/2
if (x - 2)=0=>x=2
if (5 x^2 - 2 x + 5)=0 =>x = 1/5 - (2i*sqrt(6))/5 or x = 1/5 + (2i*sqrt(6))/5

Cross-multiplication is a very helpful step one.