Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "38.665" was replaced by "(38665/1000)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            7733
 Simplify   ————
            200 

Equation at the end of step  1  :

       49       3479   7733
  (0-((——•(x2))•————))-————  = 0 
       10       100    200 

Step  2  :

            3479
 Simplify   ————
            100 

Equation at the end of step  2  :

       49       3479   7733
  (0-((——•(x2))•————))-————  = 0 
       10       100    200 
 Step  3  :
            49
 Simplify   ——
            10

Equation at the end of step  3  :

          49         3479      7733
  (0 -  ((—— • x2) • ————)) -  ————  = 0 
          10         100       200 

Step  4  :

Equation at the end of step  4  :

         49x2   3479      7733
  (0 -  (———— • ————)) -  ————  = 0 
          10    100       200 

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       1000 

      The right denominator is :       200 

      Least Common Multiple:
      1000 

Calculating Multipliers :

 5.2    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 = 1

   Right_M = L.C.M / R_Deno = 5

Making Equivalent Fractions :

 5.3      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.      -170471x2
   ——————————————————  =   —————————
         L.C.M               1000   

   R. Mult. • R. Num.      7733 • 5
   ——————————————————  =   ————————
         L.C.M               1000  

Adding fractions that have a common denominator :

 5.4       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:

 -170471x2 - (7733 • 5)     -170471x2 - 38665
 ——————————————————————  =  —————————————————
          1000                    1000       

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -170471x² - 38665  =   -1 • (170471x² + 38665) 

Trying to factor as a Difference of Squares :

 6.2      Factoring:  170471x² + 38665 

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 :  170471  is not a square !!

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

Equation at the end of step  6  :

  -170471x2 - 38665
  —————————————————  = 0 
        1000       

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:

  -170471x2-38665
  ——————————————— • 1000 = 0 • 1000
       1000      

Now, on the left hand side, the  1000  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -170471x²-38665  = 0

Solving a Single Variable Equation :

 7.2      Solve  :    -170471x²-38665 = 0 

 Add  38665  to both sides of the equation : 
                      -170471x² = 38665
Multiply both sides of the equation by (-1) :  170471x² = -38665


Divide both sides of the equation by 170471:
                     x² = -38665/170471 = -0.227
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -38665/170471  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -38665/170471  =
                    √ -1• 38665/170471   =
                    √ -1 •√  38665/170471   =
                    i •  √ 38665/170471

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x=  0.0000 + 0.4762 i 
                      x=  0.0000 - 0.4762 i 

Two solutions were found :

1.   x=  0.0000 - 0.4762 i 
   
2.   x=  0.0000 + 0.4762 i