Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.797448" was replaced by "(797448/1000000)".

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                1/(1+r)^3-((797448/1000000))=0 

Step by step solution :

Step  1  :

             99681
 Simplify   ——————
            125000

Equation at the end of step  1  :

       1         99681
  —————————— -  ——————  = 0 
  ((r + 1)3)    125000

Step  2  :

                1   
 Simplify   ————————
            (r + 1)3

Equation at the end of step  2  :

      1        99681
  ———————— -  ——————  = 0 
  (r + 1)3    125000

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       (r+1)³  

      The right denominator is :       125000 

      Least Common Multiple:
      125000 • (r+1)³  

Calculating Multipliers :

 3.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 125000

   Right_M = L.C.M / R_Deno = (r+1)³

Making Equivalent Fractions :

 3.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.           125000    
   ——————————————————  =   ———————————————
         L.C.M             125000 • (r+1)3

   R. Mult. • R. Num.       99681 • (r+1)3
   ——————————————————  =   ———————————————
         L.C.M             125000 • (r+1)3

Adding fractions that have a common denominator :

 3.4       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 125000 - (99681 • (r+1)3)     -99681r3 - 299043r2 - 299043r + 25319
 —————————————————————————  =  —————————————————————————————————————
      125000 • (r+1)3              125000 • (r3 + 3r2 + 3r + 1)     

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   -99681r³ - 299043r² - 299043r + 25319  = 

  -1 • (99681r³ + 299043r² + 299043r - 25319) 

Checking for a perfect cube :

 4.2    99681r³ + 299043r² + 299043r - 25319  is not a perfect cube

Trying to factor by pulling out :

 4.3      Factoring:  99681r³ + 299043r² + 299043r - 25319 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  299043r - 25319 
Group 2:  299043r² + 99681r³ 

Pull out from each group separately :

Group 1:   (299043r - 25319) • (1)
Group 2:   (r + 3) • (99681r²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Checking for a perfect cube :

 4.4    Factoring:  r³ + 3r² + 3r + 1 
 .

  r³ + 3r² + 3r + 1  is a perfect cube which means it is the cube of another polynomial 

 In our case, the cubic root of  r³ + 3r² + 3r + 1  is  r + 1  

 Factorization is  (r + 1)³

Polynomial Roots Calculator :

 4.5    Find roots (zeroes) of :       F(r) = 99681r³ + 299043r² + 299043r - 25319
Polynomial Roots Calculator is a set of methods aimed at finding values of  r  for which   F(r)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  r  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  99681  and the Trailing Constant is  -25319.

 The factor(s) are:

of the Leading Coefficient :  1,3 ,149 ,223 ,447 ,669 ,33227 ,99681
 of the Trailing Constant :  1 ,7 ,3617 ,25319

 Let us test ....

Note - For tidiness, printing of 59 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  4  :

  -99681r3 - 299043r2 - 299043r + 25319
  —————————————————————————————————————  = 0 
            125000 • (r + 1)3          

Step  5  :

When a fraction equals zero :

 5.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  -99681r3-299043r2-299043r+25319
  ——————————————————————————————— • 125000•(r+1)3 = 0 • 125000•(r+1)3
           125000•(r+1)3         

Now, on the left hand side, the  125000 • (r+1)³   cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -99681r³-299043r²-299043r+25319  = 0

Cubic Equations :

 5.2     Solve   -99681r³-299043r²-299043r+25319 = 0

Future releases of Tiger-Algebra will solve equations of the third degree directly.

Meanwhile we will use the Bisection method to approximate one real solution.

Approximating a root using the Bisection Method :

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is   F(r) = -99681r³ - 299043r² - 299043r + 25319

At   r=   1.00   F(r)  is equal to  -672448.00 
At   r=   0.00   F(r)  is equal to  25319.00 

Intuitively we feel, and justly so, that since  F(r)  is negative on one side of the interval, and positive on the other side then, somewhere inside this interval,  F(r)  is zero

Procedure :
(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

    Left       Value(Left)     Right       Value(Right)

 1.000000000    -672448.00  0.000000000      25319.00
 1.000000000    -672448.00  0.000000000      25319.00
 0.500000000    -211423.38  0.000000000      25319.00
 0.250000000     -69689.45  0.000000000      25319.00
 0.125000000     -16928.61  0.000000000      25319.00
 0.125000000     -16928.61  0.062500000       5436.340
 0.093750000      -5426.723  0.062500000       5436.340
 0.093750000      -5426.723  0.078125000   83.520397186
 0.085937500      -2651.781  0.078125000   83.520397186
 0.082031250      -1279.193  0.078125000   83.520397186
 0.080078125 -596.604136653  0.078125000   83.520397186
 0.079101562 -256.234121160  0.078125000   83.520397186
 0.078613281  -86.279959656  0.078125000   83.520397186
 0.078369141   -1.360560004  0.078125000   83.520397186
 0.078369141   -1.360560004  0.078247070   41.084723355
 0.078369141   -1.360560004  0.078308105   19.863282934
 0.078369141   -1.360560004  0.078338623    9.251661788
 0.078369141   -1.360560004  0.078353882    3.945625974
 0.078369141   -1.360560004  0.078361511    1.292551756
 0.078365326   -0.033999432  0.078361511    1.292551756
 0.078365326   -0.033999432  0.078363419    0.629277335
 0.078365326   -0.033999432  0.078364372    0.297639245
 0.078365326   -0.033999432  0.078364849    0.131819980
 0.078365326   -0.033999432  0.078365088    0.048910293
 0.078365326   -0.033999432  0.078365207    0.007455435
 0.078365266   -0.013271997  0.078365207    0.007455435
 0.078365237   -0.002908281  0.078365207    0.007455435
 0.078365237   -0.002908281  0.078365222    0.002273577
 0.078365229   -0.000317352  0.078365222    0.002273577
 0.078365229   -0.000317352  0.078365225    0.000978113
 0.078365229   -0.000317352  0.078365227    0.000330380
 0.078365229   -0.000317352  0.078365228    0.000006514
 0.078365229   -0.000155419  0.078365228    0.000006514
 0.078365228   -0.000074452  0.078365228    0.000006514
 0.078365228   -0.000033969  0.078365228    0.000006514
 0.078365228   -0.000013727  0.078365228    0.000006514
 0.078365228   -0.000003606  0.078365228    0.000006514
 0.078365228   -0.000003606  0.078365228    0.000001454

     Next Middle will get us close enough to zero:

     F(  0.078365228 ) is   0.000000189  

     The desired approximation of the solution is:

       r ≓ 0.078365228

     Note, ≓ is the approximation symbol

One solution was found :