Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "3.2" was replaced by "(32/10)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            16
 Simplify   ——
            5 

Equation at the end of step  1  :

         4          16
  ((0-(————•(x2)))+(——•x))-200  = 0 
       1000         5 
 Step  2  :
             1 
 Simplify   ———
            250

Equation at the end of step  2  :

           1            16x     
  ((0 -  (——— • x2)) +  ———) -  200  = 0 
          250            5      

Step  3  :

Equation at the end of step  3  :

          x2     16x     
  ((0 -  ———) +  ———) -  200  = 0 
         250      5      

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       250 

      The right denominator is :       5 

      Least Common Multiple:
      250 

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 = 50

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.      -x2
   ——————————————————  =   ———
         L.C.M             250

   R. Mult. • R. Num.      16x • 50
   ——————————————————  =   ————————
         L.C.M               250   

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:

 -x2 + 16x • 50     800x - x2
 ——————————————  =  —————————
      250              250   

Equation at the end of step  4  :

  (800x - x2)    
  ——————————— -  200  = 0 
      250        

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  250  as the denominator :

           200     200 • 250
    200 =  ———  =  —————————
            1         250   

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 :

   800x - x²  =   -x • (x - 800) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -x • (x-800) - (200 • 250)     -x2 + 800x - 50000
 ——————————————————————————  =  ——————————————————
            250                        250        

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -x² + 800x - 50000  =   -1 • (x² - 800x + 50000) 

Trying to factor by splitting the middle term

 7.2     Factoring  x² - 800x + 50000 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -800x  its coefficient is  -800 .
The last term, "the constant", is  +50000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 50000 = 50000 

Step-2 : Find two factors of  50000  whose sum equals the coefficient of the middle term, which is   -800 .

For tidiness, printing of 54 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  :

  -x2 + 800x - 50000
  ——————————————————  = 0 
         250        

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:

  -x2+800x-50000
  —————————————— • 250 = 0 • 250
       250      

Now, on the left hand side, the  250  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -x²+800x-50000  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -x²+800x-50000

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, -1 , 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  400.0000  

 Plugging into the parabola formula  400.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 400.00 * 400.00 + 800.0 * 400.00 - 50000.0
or   y = 110000.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+800x-50000
Axis of Symmetry (dashed)  {x}={400.00} 
Vertex at  {x,y} = {400.00,110000.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {731.66, 0.00} 
Root 2 at  {x,y} = {68.34, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   -x²+800x-50000 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-800x+50000 = 0  Subtract  50000  from both side of the equation :
   x²-800x = -50000

Now the clever bit: Take the coefficient of  x , which is  800 , divide by two, giving  400 , and finally square it giving  160000 

Add  160000  to both sides of the equation :
  On the right hand side we have :
   -50000  +  160000    or,  (-50000/1)+(160000/1) 
  The common denominator of the two fractions is  1   Adding  (-50000/1)+(160000/1)  gives  110000/1 
  So adding to both sides we finally get :
   x²-800x+160000 = 110000

Adding  160000  has completed the left hand side into a perfect square :
   x²-800x+160000  =
   (x-400) • (x-400)  =
  (x-400)²
Things which are equal to the same thing are also equal to one another. Since
   x²-800x+160000 = 110000 and
   x²-800x+160000 = (x-400)²
then, according to the law of transitivity,
   (x-400)² = 110000

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-400)²   is
   (x-400)^2/2 =
  (x-400)¹ =
   x-400

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-400 = √ 110000

Add  400  to both sides to obtain:
   x = 400 + √ 110000

Since a square root has two values, one positive and the other negative
   x² - 800x + 50000 = 0
   has two solutions:
  x = 400 + √ 110000
   or
  x = 400 - √ 110000

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    -x²+800x-50000 = 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   =     -1
                      B   =   800
                      C   =  -50000

Accordingly,  B²  -  4AC   =
                     640000 - 200000 =
                     440000

Applying the quadratic formula :

               -800 ± √ 440000
   x  =    —————————
                          -2

Can  √ 440000 be simplified ?

Yes!   The prime factorization of  440000   is
   2•2•2•2•2•2•5•5•5•5•11 
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).

√ 440000   =  √ 2•2•2•2•2•2•5•5•5•5•11   =2•2•2•5•5•√ 11   =
                ±  200 • √ 11

  √ 11   , rounded to 4 decimal digits, is   3.3166
 So now we are looking at:
           x  =  ( -800 ± 200 •  3.317 ) / -2

Two real solutions:

 x =(-800+√440000)/-2=400-100√ 11 = 68.338

or:

 x =(-800-√440000)/-2=400+100√ 11 = 731.662

Two solutions were found :

1.  x =(-800-√440000)/-2=400+100√ 11 = 731.662
2.  x =(-800+√440000)/-2=400-100√ 11 = 68.338