Solution - Adding, subtracting and finding the least common multiple

Rearrange:

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

                     -9/22*x-7/6-(x^3)=0 

Step by step solution :

Step  1  :

            7
 Simplify   —
            6

Equation at the end of step  1  :

           9          7     
  ((0 -  (—— • x)) -  —) -  x3  = 0 
          22          6     

Step  2  :

             9
 Simplify   ——
            22

Equation at the end of step  2  :

           9          7     
  ((0 -  (—— • x)) -  —) -  x3  = 0 
          22          6     

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       22 

      The right denominator is :       6 

      Least Common Multiple:
      66 

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 = 3

   Right_M = L.C.M / R_Deno = 11

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.      -9x • 3
   ——————————————————  =   ———————
         L.C.M               66   

   R. Mult. • R. Num.      7 • 11
   ——————————————————  =   ——————
         L.C.M               66  

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:

 -9x • 3 - (7 • 11)     -27x - 77
 ——————————————————  =  —————————
         66                66    

Equation at the end of step  3  :

  (-27x - 77)    
  ——————————— -  x3  = 0 
      66         

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  66  as the denominator :

          x3     x3 • 66
    x3 =  ——  =  ———————
          1        66   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -27x - 77  =   -1 • (27x + 77) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 (-27x-77) - (x3 • 66)     -66x3 - 27x - 77
 —————————————————————  =  ————————————————
          66                      66       

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -66x³ - 27x - 77  =   -1 • (66x³ + 27x + 77) 

Polynomial Roots Calculator :

 6.2    Find roots (zeroes) of :       F(x) = 66x³ + 27x + 77
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  66  and the Trailing Constant is  77.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,6 ,11 ,22 ,33 ,66
 of the Trailing Constant :  1 ,7 ,11 ,77

 Let us test ....

Note - For tidiness, printing of 43 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  6  :

  -66x3 - 27x - 77
  ————————————————  = 0 
         66       

Step  7  :

When a fraction equals zero :

 7.1    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:

  -66x3-27x-77
  ———————————— • 66 = 0 • 66
       66     

Now, on the left hand side, the  66  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -66x³-27x-77  = 0

Cubic Equations :

 7.2     Solve   -66x³-27x-77 = 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(x) = -66x³ - 27x - 77

At   x=   0.00   F(x)  is equal to  -77.00 
At   x=   -1.00   F(x)  is equal to  16.00 

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

 0.000000000  -77.000000000 -1.000000000   16.000000000
 0.000000000  -77.000000000 -1.000000000   16.000000000
-0.500000000  -55.250000000 -1.000000000   16.000000000
-0.750000000  -28.906250000 -1.000000000   16.000000000
-0.875000000   -9.160156250 -1.000000000   16.000000000
-0.875000000   -9.160156250 -0.937500000    2.694824219
-0.906250000   -3.407897949 -0.937500000    2.694824219
-0.921875000   -0.401100159 -0.937500000    2.694824219
-0.921875000   -0.401100159 -0.929687500    1.135626793
-0.921875000   -0.401100159 -0.925781250    0.364466310
-0.923828125   -0.019014701 -0.925781250    0.364466310
-0.923828125   -0.019014701 -0.924804688    0.172551176
-0.923828125   -0.019014701 -0.924316406    0.076724603
-0.923828125   -0.019014701 -0.924072266    0.028844045
-0.923828125   -0.019014701 -0.923950195    0.004911946
-0.923889160   -0.007052059 -0.923950195    0.004911946
-0.923919678   -0.001070227 -0.923950195    0.004911946
-0.923919678   -0.001070227 -0.923934937    0.001920817
-0.923919678   -0.001070227 -0.923927307    0.000425284
-0.923923492   -0.000322474 -0.923927307    0.000425284
-0.923923492   -0.000322474 -0.923925400    0.000051405
-0.923924446   -0.000135535 -0.923925400    0.000051405
-0.923924923   -0.000042065 -0.923925400    0.000051405
-0.923924923   -0.000042065 -0.923925161    0.000004670
-0.923925042   -0.000018698 -0.923925161    0.000004670
-0.923925102   -0.000007014 -0.923925161    0.000004670
-0.923925132   -0.000001172 -0.923925161    0.000004670

     Next Middle will get us close enough to zero:

     F( -0.923925139 ) is   0.000000288  

     The desired approximation of the solution is:

       x ≓ -0.923925139

     Note, ≓ is the approximation symbol

One solution was found :