Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.02565" was replaced by "(02565/100000)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

             513 
 Simplify   —————
            20000

Equation at the end of step  1  :

              513      513 
  ((4•(x2))+(————•x))-—————  = 0 
             1000     20000

Step  2  :

             513
 Simplify   ————
            1000

Equation at the end of step  2  :

                   513           513 
  ((4 • (x2)) +  (———— • x)) -  —————  = 0 
                  1000          20000
 Step  3  :

Equation at the end of step  3  :

           513x      513 
  (22x2 +  ————) -  —————  = 0 
           1000     20000

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  1000  as the denominator :

             22x2     22x2 • 1000
     22x2 =  ————  =  ———————————
              1          1000    

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 :

 4.2       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:

 22x2 • 1000 + 513x     4000x2 + 513x
 ——————————————————  =  —————————————
        1000                1000     

Equation at the end of step  4  :

  (4000x2 + 513x)     513 
  ——————————————— -  —————  = 0 
       1000          20000

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   4000x² + 513x  =   x • (4000x + 513) 

Calculating the Least Common Multiple :

 6.2    Find the Least Common Multiple

      The left denominator is :       1000 

      The right denominator is :       20000 

      Least Common Multiple:
      20000 

Calculating Multipliers :

 6.3    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 = 20

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 6.4      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 • (4000x+513) • 20
   ——————————————————  =   ————————————————————
         L.C.M                    20000        

   R. Mult. • R. Num.       513 
   ——————————————————  =   —————
         L.C.M             20000

Adding fractions that have a common denominator :

 6.5       Adding up the two equivalent fractions

 x • (4000x+513) • 20 - (513)     80000x2 + 10260x - 513
 ————————————————————————————  =  ——————————————————————
            20000                         20000         

Trying to factor by splitting the middle term

 6.6     Factoring  80000x² + 10260x - 513 

The first term is,  80000x²  its coefficient is  80000 .
The middle term is,  +10260x  its coefficient is  10260 .
The last term, "the constant", is  -513 

Step-1 : Multiply the coefficient of the first term by the constant   80000 • -513 = -41040000 

Step-2 : Find two factors of  -41040000  whose sum equals the coefficient of the middle term, which is   10260 .


Numbers too big. Method shall not be applied

Equation at the end of step  6  :

  80000x2 + 10260x - 513
  ——————————————————————  = 0 
          20000         

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:

  80000x2+10260x-513
  —————————————————— • 20000 = 0 • 20000
        20000       

Now, on the left hand side, the  20000  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   80000x²+10260x-513  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = 80000x²+10260x-513

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, 80000 , 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  -0.0641  

 Plugging into the parabola formula  -0.0641  for  x  we can calculate the  y -coordinate : 
  y = 80000.0 * -0.06 * -0.06 + 10260.0 * -0.06 - 513.0
or   y = -841.961

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 80000x²+10260x-513
Axis of Symmetry (dashed)  {x}={-0.06} 
Vertex at  {x,y} = {-0.06,-841.96} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.17, 0.00} 
Root 2 at  {x,y} = { 0.04, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 7.3     Solving    80000x²+10260x-513 = 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   =     80000.00
     B   =    10260.00
     C   =    -513.00

   B² = 105267600.00 
   4AC = -164160000.00 
   B² - 4AC = 269427600.00 
   SQRT(B²-4AC) =  16414.25
   x=( -10260.00 ± 16414.25) / 160000.00 
   x =  0.03846
   x =  -0.16671

Two solutions were found :

1.    x =  -0.16671
   
2.    x =  0.03846