Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): ".0546" was replaced by "(0546/10000)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

             273
 Simplify   ————
            5000

Equation at the end of step  1  :

        475          339      273
  ((0-(————•(x2)))+(————•x))+————  = 0 
       1000         1000     5000

Step  2  :

             339
 Simplify   ————
            1000

Equation at the end of step  2  :

        475          339      273
  ((0-(————•(x2)))+(————•x))+————  = 0 
       1000         1000     5000
 Step  3  :
            19
 Simplify   ——
            40

Equation at the end of step  3  :

          19           339x      273
  ((0 -  (—— • x2)) +  ————) +  ————  = 0 
          40           1000     5000

Step  4  :

Equation at the end of step  4  :

         19x2     339x      273
  ((0 -  ————) +  ————) +  ————  = 0 
          40      1000     5000

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       40 

      The right denominator is :       1000 

      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 = 25

   Right_M = L.C.M / R_Deno = 1

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.      -19x2 • 25
   ——————————————————  =   ——————————
         L.C.M                1000   

   R. Mult. • R. Num.      339x
   ——————————————————  =   ————
         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:

 -19x2 • 25 + 339x     339x - 475x2
 —————————————————  =  ————————————
       1000                1000    

Equation at the end of step  5  :

  (339x - 475x2)     273
  —————————————— +  ————  = 0 
       1000         5000

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   339x - 475x²  =   -x • (475x - 339) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       1000 

      The right denominator is :       5000 

      Least Common Multiple:
      5000 

Calculating Multipliers :

 7.3    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 = 5

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      -x • (475x-339) • 5
   ——————————————————  =   ———————————————————
         L.C.M                    5000        

   R. Mult. • R. Num.       273
   ——————————————————  =   ————
         L.C.M             5000

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 -x • (475x-339) • 5 + 273     -2375x2 + 1695x + 273
 —————————————————————————  =  —————————————————————
           5000                        5000         

Trying to factor by splitting the middle term

 7.6     Factoring  -2375x² + 1695x + 273 

The first term is,  -2375x²  its coefficient is  -2375 .
The middle term is,  +1695x  its coefficient is  1695 .
The last term, "the constant", is  +273 

Step-1 : Multiply the coefficient of the first term by the constant   -2375 • 273 = -648375 

Step-2 : Find two factors of  -648375  whose sum equals the coefficient of the middle term, which is   1695 .

For tidiness, printing of 58 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  7  :

  -2375x2 + 1695x + 273
  —————————————————————  = 0 
          5000         

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:

  -2375x2+1695x+273
  ————————————————— • 5000 = 0 • 5000
        5000       

Now, on the left hand side, the  5000  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -2375x²+1695x+273  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -2375x²+1695x+273

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, -2375 , 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   0.3568  

 Plugging into the parabola formula   0.3568  for  x  we can calculate the  y -coordinate : 
  y = -2375.0 * 0.36 * 0.36 + 1695.0 * 0.36 + 273.0
or   y = 575.424

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -2375x²+1695x+273
Axis of Symmetry (dashed)  {x}={ 0.36} 
Vertex at  {x,y} = { 0.36,575.42} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.85, 0.00} 
Root 2 at  {x,y} = {-0.14, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   -2375x²+1695x+273 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 2375x²-1695x-273 = 0  Divide both sides of the equation by  2375  to have 1 as the coefficient of the first term :
   x²-(339/475)x-(273/2375) = 0

Add  273/2375  to both side of the equation :
   x²-(339/475)x = 273/2375

Now the clever bit: Take the coefficient of  x , which is  339/475 , divide by two, giving  339/950 , and finally square it giving  339/950 

Add  339/950  to both sides of the equation :
  On the right hand side we have :
   273/2375  +  339/950   The common denominator of the two fractions is  4750   Adding  (546/4750)+(1695/4750)  gives  2241/4750 
  So adding to both sides we finally get :
   x²-(339/475)x+(339/950) = 2241/4750

Adding  339/950  has completed the left hand side into a perfect square :
   x²-(339/475)x+(339/950)  =
   (x-(339/950)) • (x-(339/950))  =
  (x-(339/950))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(339/475)x+(339/950) = 2241/4750 and
   x²-(339/475)x+(339/950) = (x-(339/950))²
then, according to the law of transitivity,
   (x-(339/950))² = 2241/4750

We'll refer to this Equation as  Eq. #8.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(339/950))²   is
   (x-(339/950))^2/2 =
  (x-(339/950))¹ =
   x-(339/950)

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-(339/950) = √ 2241/4750

Add  339/950  to both sides to obtain:
   x = 339/950 + √ 2241/4750

Since a square root has two values, one positive and the other negative
   x² - (339/475)x - (273/2375) = 0
   has two solutions:
  x = 339/950 + √ 2241/4750
   or
  x = 339/950 - √ 2241/4750

Note that  √ 2241/4750 can be written as
  √ 2241  / √ 4750 

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    -2375x²+1695x+273 = 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   =    -2375
                      B   =   1695
                      C   =  273

Accordingly,  B²  -  4AC   =
                     2873025 - (-2593500) =
                     5466525

Applying the quadratic formula :

               -1695 ± √ 5466525
   x  =    ——————————
                           -4750

Can  √ 5466525 be simplified ?

Yes!   The prime factorization of  5466525   is
   3•5•5•23•3169 
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).

√ 5466525   =  √ 3•5•5•23•3169   =
                ±  5 • √ 218661

  √ 218661   , rounded to 4 decimal digits, is  467.6120
 So now we are looking at:
           x  =  ( -1695 ± 5 •  467.612 ) / -4750

Two real solutions:

 x =(-1695+√5466525)/-4750=(339-√ 218661 )/950= -0.135

or:

 x =(-1695-√5466525)/-4750=(339+√ 218661 )/950= 0.849

Two solutions were found :

1.  x =(-1695-√5466525)/-4750=(339+√ 218661 )/950= 0.849
2.  x =(-1695+√5466525)/-4750=(339-√ 218661 )/950= -0.135