Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "759.8" was replaced by "(7598/10)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            3799
 Simplify   ————
             5  

Equation at the end of step  1  :

       2027         3799
  ((0-(————•(x2)))+(————•x))-3146  = 0 
       100           5  
 Step  2  :
            2027
 Simplify   ————
            100 

Equation at the end of step  2  :

          2027           3799x     
  ((0 -  (———— • x2)) +  —————) -  3146  = 0 
          100              5       

Step  3  :

Equation at the end of step  3  :

         2027x2     3799x     
  ((0 -  ——————) +  —————) -  3146  = 0 
          100         5       

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       100 

      The right denominator is :       5 

      Least Common Multiple:
      100 

Calculating Multipliers :

 4.2    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 = 20

Making Equivalent Fractions :

 4.3      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.      -2027x2
   ——————————————————  =   ———————
         L.C.M               100  

   R. Mult. • R. Num.      3799x • 20
   ——————————————————  =   ——————————
         L.C.M                100    

Adding fractions that have a common denominator :

 4.4       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:

 -2027x2 + 3799x • 20     75980x - 2027x2
 ————————————————————  =  ———————————————
         100                    100      

Equation at the end of step  4  :

  (75980x - 2027x2)    
  ————————————————— -  3146  = 0 
         100           

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  100  as the denominator :

            3146     3146 • 100
    3146 =  ————  =  ——————————
             1          100    

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  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   75980x - 2027x²  =   -x • (2027x - 75980) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -x • (2027x-75980) - (3146 • 100)     -2027x2 + 75980x - 314600
 —————————————————————————————————  =  —————————————————————————
                100                               100           

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -2027x² + 75980x - 314600  =   -1 • (2027x² - 75980x + 314600) 

Trying to factor by splitting the middle term

 7.2     Factoring  2027x² - 75980x + 314600 

The first term is,  2027x²  its coefficient is  2027 .
The middle term is,  -75980x  its coefficient is  -75980 .
The last term, "the constant", is  +314600 

Step-1 : Multiply the coefficient of the first term by the constant   2027 • 314600 = 637694200 

Step-2 : Find two factors of  637694200  whose sum equals the coefficient of the middle term, which is   -75980 .


Numbers too big. Method shall not be applied

Equation at the end of step  7  :

  -2027x2 + 75980x - 314600
  —————————————————————————  = 0 
             100           

Step  8  :

When a fraction equals zero :

 8.1    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:

  -2027x2+75980x-314600
  ————————————————————— • 100 = 0 • 100
           100         

Now, on the left hand side, the  100  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -2027x²+75980x-314600  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -2027x²+75980x-314600

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -2027 , is negative (smaller 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  18.7420  

 Plugging into the parabola formula  18.7420  for  x  we can calculate the  y -coordinate : 
  y = -2027.0 * 18.74 * 18.74 + 75980.0 * 18.74 - 314600.0
or   y = 397407.943

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -2027x²+75980x-314600
Axis of Symmetry (dashed)  {x}={18.74} 
Vertex at  {x,y} = {18.74,397407.94} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {32.74, 0.00} 
Root 2 at  {x,y} = { 4.74, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 8.3     Solving    -2027x²+75980x-314600 = 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   =     -2027.00
     B   =    75980.00
     C   =    -314600.00

   B² = 5772960400.00 
   4AC = 2550776800.00 
   B² - 4AC = 3222183600.00 
   SQRT(B²-4AC) =  56764.28
   x=( -75980.00 ± 56764.28) / -4054.00 
   x =  4.73994
   x =  32.74403

Two solutions were found :

1.    x =  32.74403
   
2.    x =  4.73994