Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.0016856" was replaced by "(0016856/10000000)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

              2107 
 Simplify   ———————
            1250000

Equation at the end of step  1  :

              353976              2107 
  ((x2) -  (—————————— • x)) +  ———————  = 0 
            1000000000          1250000

Step  2  :

              44247  
 Simplify   —————————
            125000000

Equation at the end of step  2  :

              44247              2107 
  ((x2) -  (————————— • x)) +  ———————  = 0 
            125000000          1250000
 Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  125000000  as the denominator :

           x2     x2 • 125000000
     x2 =  ——  =  ——————————————
           1        125000000   

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

Adding fractions that have a common denominator :

 3.2       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:

 x2 • 125000000 - (44247x)     125000000x2 - 44247x
 —————————————————————————  =  ————————————————————
         125000000                  125000000      

Equation at the end of step  3  :

  (125000000x2 - 44247x)      2107 
  —————————————————————— +  ———————  = 0 
        125000000           1250000

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   125000000x² - 44247x  =   x • (125000000x - 44247) 

Calculating the Least Common Multiple :

 5.2    Find the Least Common Multiple

      The left denominator is :       125000000 

      The right denominator is :       1250000 

      Least Common Multiple:
      125000000 

Calculating Multipliers :

 5.3    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 = 100

Making Equivalent Fractions :

 5.4      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.      x • (125000000x-44247)
   ——————————————————  =   ——————————————————————
         L.C.M                   125000000       

   R. Mult. • R. Num.      2107 • 100
   ——————————————————  =   ——————————
         L.C.M             125000000 

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 x • (125000000x-44247) + 2107 • 100     125000000x2 - 44247x + 210700
 ———————————————————————————————————  =  —————————————————————————————
              125000000                            125000000          

Trying to factor by splitting the middle term

 5.6     Factoring  125000000x² - 44247x + 210700 

The first term is,  125000000x²  its coefficient is  125000000 .
The middle term is,  -44247x  its coefficient is  -44247 .
The last term, "the constant", is  +210700 

Step-1 : Multiply the coefficient of the first term by the constant

Numbers too big. Method shall not be applied

Equation at the end of step  5  :

  125000000x2 - 44247x + 210700
  —————————————————————————————  = 0 
            125000000          

Step  6  :

When a fraction equals zero :

 6.1    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:

  125000000x2-44247x+210700
  ————————————————————————— • 125000000 = 0 • 125000000
          125000000        

Now, on the left hand side, the  125000000  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   125000000x²-44247x+210700  = 0

Parabola, Finding the Vertex :

 6.2      Find the Vertex of   y = 125000000x²-44247x+210700

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 125000000 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   0.0002  

 Plugging into the parabola formula   0.0002  for  x  we can calculate the  y -coordinate : 
  y = 125000000.0 * 0.00 * 0.00 - 44247.0 * 0.00 + 210700.0
or   y = 210696.084

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 125000000x²-44247x+210700
Axis of Symmetry (dashed)  {x}={ 0.00} 
Vertex at  {x,y} = { 0.00,210696.08} 
Function has no real roots

Solve Quadratic Equation using the Quadratic Formula

 6.3     Solving    125000000x²-44247x+210700 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax²+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  x =   ————————
                      2A

  In our case:   
     A   =    125000000.00
     B   =    -44247.00
     C   =    210700.00

   B² = 1957797009.00 
   4AC = 105350000000000.00 
   B² - 4AC = -105348042202991.00 
   SQRT(B²-4AC) =  10263919.44  i  
  x=( 44247.00 ±10263919.44 i ) /250000000.00 
   x =  0.00018  -  0.04106 i 
   x =  0.00018  -  0.04106 i 

Two solutions were found :

1.    x =  0.00018  -  0.04106 i 
   
2.    x =  0.00018  -  0.04106 i