Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "1.06" was replaced by "(106/100)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            53
 Simplify   ——
            50

Equation at the end of step  1  :

         2          53
  ((0-(————•(x2)))+(——•x))-52  = 0 
       1000         50
 Step  2  :
             1 
 Simplify   ———
            500

Equation at the end of step  2  :

           1            53x     
  ((0 -  (——— • x2)) +  ———) -  52  = 0 
          500           50      

Step  3  :

Equation at the end of step  3  :

          x2     53x     
  ((0 -  ———) +  ———) -  52  = 0 
         500     50      

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       500 

      The right denominator is :       50 

      Least Common Multiple:
      500 

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 = 10

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             500

   R. Mult. • R. Num.      53x • 10
   ——————————————————  =   ————————
         L.C.M               500   

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 + 53x • 10     530x - x2
 ——————————————  =  —————————
      500              500   

Equation at the end of step  4  :

  (530x - x2)    
  ——————————— -  52  = 0 
      500        

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  500  as the denominator :

          52     52 • 500
    52 =  ——  =  ————————
          1        500   

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 :

   530x - x²  =   -x • (x - 530) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -x • (x-530) - (52 • 500)     -x2 + 530x - 26000
 —————————————————————————  =  ——————————————————
            500                       500        

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -x² + 530x - 26000  =   -1 • (x² - 530x + 26000) 

Trying to factor by splitting the middle term

 7.2     Factoring  x² - 530x + 26000 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -530x  its coefficient is  -530 .
The last term, "the constant", is  +26000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 26000 = 26000 

Step-2 : Find two factors of  26000  whose sum equals the coefficient of the middle term, which is   -530 .

For tidiness, printing of 74 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 + 530x - 26000
  ——————————————————  = 0 
         500        

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+530x-26000
  —————————————— • 500 = 0 • 500
       500      

Now, on the left hand side, the  500  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -x²+530x-26000  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -x²+530x-26000

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  265.0000  

 Plugging into the parabola formula  265.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 265.00 * 265.00 + 530.0 * 265.00 - 26000.0
or   y = 44225.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+530x-26000
Axis of Symmetry (dashed)  {x}={265.00} 
Vertex at  {x,y} = {265.00,44225.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {475.30, 0.00} 
Root 2 at  {x,y} = {54.70, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   -x²+530x-26000 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-530x+26000 = 0  Subtract  26000  from both side of the equation :
   x²-530x = -26000

Now the clever bit: Take the coefficient of  x , which is  530 , divide by two, giving  265 , and finally square it giving  70225 

Add  70225  to both sides of the equation :
  On the right hand side we have :
   -26000  +  70225    or,  (-26000/1)+(70225/1) 
  The common denominator of the two fractions is  1   Adding  (-26000/1)+(70225/1)  gives  44225/1 
  So adding to both sides we finally get :
   x²-530x+70225 = 44225

Adding  70225  has completed the left hand side into a perfect square :
   x²-530x+70225  =
   (x-265) • (x-265)  =
  (x-265)²
Things which are equal to the same thing are also equal to one another. Since
   x²-530x+70225 = 44225 and
   x²-530x+70225 = (x-265)²
then, according to the law of transitivity,
   (x-265)² = 44225

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-265)²   is
   (x-265)^2/2 =
  (x-265)¹ =
   x-265

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-265 = √ 44225

Add  265  to both sides to obtain:
   x = 265 + √ 44225

Since a square root has two values, one positive and the other negative
   x² - 530x + 26000 = 0
   has two solutions:
  x = 265 + √ 44225
   or
  x = 265 - √ 44225

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    -x²+530x-26000 = 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   =   530
                      C   =  -26000

Accordingly,  B²  -  4AC   =
                     280900 - 104000 =
                     176900

Applying the quadratic formula :

               -530 ± √ 176900
   x  =    —————————
                          -2

Can  √ 176900 be simplified ?

Yes!   The prime factorization of  176900   is
   2•2•5•5•29•61 
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).

√ 176900   =  √ 2•2•5•5•29•61   =2•5•√ 1769   =
                ±  10 • √ 1769

  √ 1769   , rounded to 4 decimal digits, is  42.0595
 So now we are looking at:
           x  =  ( -530 ± 10 •  42.059 ) / -2

Two real solutions:

 x =(-530+√176900)/-2=265-5√ 1769 = 54.703

or:

 x =(-530-√176900)/-2=265+5√ 1769 = 475.297

Two solutions were found :

1.  x =(-530-√176900)/-2=265+5√ 1769 = 475.297
2.  x =(-530+√176900)/-2=265-5√ 1769 = 54.703