Mister Exam

Other calculators

Expression simplification/ Fraction Decomposition into the simple/ ((4x-5)/(3x+2))^2+((3x+2)/(5-4x))^2

How do you ((4x-5)/(3x+2))^2+((3x+2)/(5-4x))^2 in partial fractions?

An expression to simplify:

The solution

You have entered [src] 
         2            2
/4*x - 5\    /3*x + 2\ 
|-------|  + |-------| 
\3*x + 2/    \5 - 4*x/
$$\left(\frac{3 x + 2}{5 - 4 x}\right)^{2} + \left(\frac{4 x - 5}{3 x + 2}\right)^{2}$$ 
((4*x - 5)/(3*x + 2))^2 + ((3*x + 2)/(5 - 4*x))^2
Fraction decomposition [src] 
337/144 - 184/(9*(2 + 3*x)) + 69/(8*(-5 + 4*x)) + 529/(9*(2 + 3*x)^2) + 529/(16*(-5 + 4*x)^2)
$$\frac{337}{144} + \frac{69}{8 \left(4 x - 5\right)} + \frac{529}{16 \left(4 x - 5\right)^{2}} - \frac{184}{9 \left(3 x + 2\right)} + \frac{529}{9 \left(3 x + 2\right)^{2}}$$ 
337       184            69            529             529      
--- - ----------- + ------------ + ------------ + --------------
144   9*(2 + 3*x)   8*(-5 + 4*x)              2                2
                                   9*(2 + 3*x)    16*(-5 + 4*x)
General simplification [src] 
          4            4
(-5 + 4*x)  + (2 + 3*x) 
------------------------
           2          2 
 (-5 + 4*x) *(2 + 3*x)
$$\frac{\left(3 x + 2\right)^{4} + \left(4 x - 5\right)^{4}}{\left(3 x + 2\right)^{2} \left(4 x - 5\right)^{2}}$$ 
((-5 + 4*x)^4 + (2 + 3*x)^4)/((-5 + 4*x)^2*(2 + 3*x)^2)
Numerical answer [src] 
2.77777777777778*(-1 + 0.8*x)^2/(0.666666666666667 + x)^2 + 0.36*(0.666666666666667 + x)^2/(1 - 0.8*x)^2
2.77777777777778*(-1 + 0.8*x)^2/(0.666666666666667 + x)^2 + 0.36*(0.666666666666667 + x)^2/(1 - 0.8*x)^2
Trigonometric part [src] 
          2            2
(-5 + 4*x)    (2 + 3*x) 
----------- + ----------
          2            2
 (2 + 3*x)    (5 - 4*x)
$$\frac{\left(4 x - 5\right)^{2}}{\left(3 x + 2\right)^{2}} + \frac{\left(3 x + 2\right)^{2}}{\left(5 - 4 x\right)^{2}}$$ 
(-5 + 4*x)^2/(2 + 3*x)^2 + (2 + 3*x)^2/(5 - 4*x)^2
Combining rational expressions [src] 
         4             2          2
(2 + 3*x)  + (-5 + 4*x) *(5 - 4*x) 
-----------------------------------
                2          2       
       (2 + 3*x) *(5 - 4*x)
$$\frac{\left(5 - 4 x\right)^{2} \left(4 x - 5\right)^{2} + \left(3 x + 2\right)^{4}}{\left(5 - 4 x\right)^{2} \left(3 x + 2\right)^{2}}$$ 
((2 + 3*x)^4 + (-5 + 4*x)^2*(5 - 4*x)^2)/((2 + 3*x)^2*(5 - 4*x)^2)
Combinatorics [src] 
                     3        4         2
641 - 1904*x - 1064*x  + 337*x  + 2616*x 
-----------------------------------------
                    2          2         
          (-5 + 4*x) *(2 + 3*x)
$$\frac{337 x^{4} - 1064 x^{3} + 2616 x^{2} - 1904 x + 641}{\left(3 x + 2\right)^{2} \left(4 x - 5\right)^{2}}$$ 
(641 - 1904*x - 1064*x^3 + 337*x^4 + 2616*x^2)/((-5 + 4*x)^2*(2 + 3*x)^2)
Common denominator [src] 
                           2          3               
337       -58604 - 441071*x  + 96600*x  + 321356*x    
--- - ------------------------------------------------
144                  2          3                    4
      14400 - 27504*x  - 24192*x  + 20160*x + 20736*x
$$- \frac{96600 x^{3} - 441071 x^{2} + 321356 x - 58604}{20736 x^{4} - 24192 x^{3} - 27504 x^{2} + 20160 x + 14400} + \frac{337}{144}$$ 
337/144 - (-58604 - 441071*x^2 + 96600*x^3 + 321356*x)/(14400 - 27504*x^2 - 24192*x^3 + 20160*x + 20736*x^4)
Assemble expression [src] 
          2            2
(-5 + 4*x)    (2 + 3*x) 
----------- + ----------
          2            2
 (2 + 3*x)    (5 - 4*x)
$$\frac{\left(4 x - 5\right)^{2}}{\left(3 x + 2\right)^{2}} + \frac{\left(3 x + 2\right)^{2}}{\left(5 - 4 x\right)^{2}}$$ 
(-5 + 4*x)^2/(2 + 3*x)^2 + (2 + 3*x)^2/(5 - 4*x)^2
Powers [src] 
          2            2
(-5 + 4*x)    (2 + 3*x) 
----------- + ----------
          2            2
 (2 + 3*x)    (5 - 4*x)
$$\frac{\left(4 x - 5\right)^{2}}{\left(3 x + 2\right)^{2}} + \frac{\left(3 x + 2\right)^{2}}{\left(5 - 4 x\right)^{2}}$$ 
(-5 + 4*x)^2/(2 + 3*x)^2 + (2 + 3*x)^2/(5 - 4*x)^2
Rational denominator [src] 
         4             2          2
(2 + 3*x)  + (-5 + 4*x) *(5 - 4*x) 
-----------------------------------
                2          2       
       (2 + 3*x) *(5 - 4*x)
$$\frac{\left(5 - 4 x\right)^{2} \left(4 x - 5\right)^{2} + \left(3 x + 2\right)^{4}}{\left(5 - 4 x\right)^{2} \left(3 x + 2\right)^{2}}$$ 
((2 + 3*x)^4 + (-5 + 4*x)^2*(5 - 4*x)^2)/((2 + 3*x)^2*(5 - 4*x)^2)
Expand expression [src] 
         2            2
(3*x + 2)    (4*x - 5) 
---------- + ----------
         2            2
(5 - 4*x)    (3*x + 2)
$$\frac{\left(4 x - 5\right)^{2}}{\left(3 x + 2\right)^{2}} + \frac{\left(3 x + 2\right)^{2}}{\left(5 - 4 x\right)^{2}}$$ 
(3*x + 2)^2/(5 - 4*x)^2 + (4*x - 5)^2/(3*x + 2)^2
    © Mister Exam – Calculator