Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-2" was replaced by "^(-2)".

Rearrange:

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

                     8*x^(-2)/3-(2/9)=0 

Step by step solution :

Step  1  :

            2
 Simplify   —
            9

Equation at the end of step  1  :

       (x-2)     2
  (8 • —————) -  —  = 0 
         3       9
 Step  2  :
            x(-2) 
 Simplify   —————
              3  

Equation at the end of step  2  :

        1      2
  (8 • ———) -  —  = 0 
       3x2     9

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       3x² 

      The right denominator is :       9 

      Least Common Multiple:
      9x² 

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 = x²

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.      8 • 3
   ——————————————————  =   —————
         L.C.M              9x2 

   R. Mult. • R. Num.      2 • x2
   ——————————————————  =   ——————
         L.C.M              9x2  

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:

 8 • 3 - (2 • x2)     24 - 2x2
 ————————————————  =  ————————
       9x2              9x2   

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   24 - 2x²  =   -2 • (x² - 12) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x² - 12 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 12 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step  4  :

  -2 • (x2 - 12)
  ——————————————  = 0 
       9x2      

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:

  -2•(x2-12)
  —————————— • 9x2 = 0 • 9x2
     9x2    

Now, on the left hand side, the  9x²  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -2  •  (x²-12)  = 0

Equations which are never true :

 5.2      Solve :    -2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 5.3      Solve  :    x²-12 = 0 

 Add  12  to both sides of the equation : 
                      x² = 12
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 12  

 Can  √ 12 be simplified ?

Yes!   The prime factorization of  12   is
   2•2•3 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 12   =  √ 2•2•3   =
                ±  2 • √ 3

The equation has two real solutions  
 These solutions are  x = 2 • ± √3 = ± 3.4641  
 

Two solutions were found :