Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.01" was replaced by "(01/100)". 2 more similar replacement(s)

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                ((2/10)-x)^2/(2*x)-((1/100))=0 

Step by step solution :

Step  1  :

             1 
 Simplify   ———
            100

Equation at the end of step  1  :

  (2                 1 
  —— - x)2) ÷ 2x -  ———  = 0 
  10                100

Step  2  :

            1
 Simplify   —
            5

Equation at the end of step  2  :

  1                 1 
  — - x)2) ÷ 2x -  ———  = 0 
  5                100

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5  as the denominator :

         x     x • 5
    x =  —  =  —————
         1       5  

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 :

 3.2       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:

 1 - (x • 5)     1 - 5x
 ———————————  =  ——————
      5            5   

Equation at the end of step  3  :

  (1 - 5x)             1 
 (————————)2) ÷ 2x -  ———  = 0 
     5                100

Step  4  :

Equation at the end of step  4  :

  (1 - 5x)2          1 
  ————————— ÷ 2x -  ———  = 0 
     52             100

Step  5  :

         (1-5x)2      
 Divide  ———————  by  2x
           52         

Equation at the end of step  5  :

  (1 - 5x)2     1 
  ————————— -  ———  = 0 
   (52•2x)     100

Step  6  :

 6.1      Finding a Common Denominator   The left  52 • 2x 
 The right  100 

 The product of any two denominators can be used
 as a common denominator. 

 Said product is not necessarily the least common 
 denominator.

 As a matter of fact, whenever the two denominators
have a common factor, their product will be bigger
 than the least common denominator. 

Anyway, the product is a fine common denominator and
 can perfectly be used for 
 calculating multipliers, as well as for generating
 equivalent fractions. 

 52 • 2x • 100   will be used as a common denominator.

Calculating Multipliers :

 6.2    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 = 100

   Right_M = L.C.M / R_Deno = 5² • 2x

Making Equivalent Fractions :

 6.3      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.                  (1-5x)2 • 100
   ——————————————————  =               —————————————
             Common denominator        (52•2x) • 100

   R. Mult. • R. Num.                     (52•2x)   
   ——————————————————  =               —————————————
             Common denominator        (52•2x) • 100

Adding fractions that have a common denominator :

 6.4       Adding up the two equivalent fractions

 (1-5x)2 • 100 - ((52•2x))     2500x2 - 1050x + 100
 —————————————————————————  =  ————————————————————
       (52•2x) • 100                50x • 100      

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   2500x² - 1050x + 100  =   50 • (50x² - 21x + 2) 

Trying to factor by splitting the middle term

 7.2     Factoring  50x² - 21x + 2 

The first term is,  50x²  its coefficient is  50 .
The middle term is,  -21x  its coefficient is  -21 .
The last term, "the constant", is  +2 

Step-1 : Multiply the coefficient of the first term by the constant   50 • 2 = 100 

Step-2 : Find two factors of  100  whose sum equals the coefficient of the middle term, which is   -21 .

For tidiness, printing of 12 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  :

  50 • (50x2 - 21x + 2)
  —————————————————————  = 0 
          5000x        

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:

  50•(50x2-21x+2)
  ——————————————— • 5000x = 0 • 5000x
       5000x     

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

The equation now takes the shape :
   50  •  (50x²-21x+2)  = 0

Equations which are never true :

 8.2      Solve :    50   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Parabola, Finding the Vertex :

 8.3      Find the Vertex of   y = 50x²-21x+2

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, 50 , 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   0.2100  

 Plugging into the parabola formula   0.2100  for  x  we can calculate the  y -coordinate : 
  y = 50.0 * 0.21 * 0.21 - 21.0 * 0.21 + 2.0
or   y = -0.205

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 50x²-21x+2
Axis of Symmetry (dashed)  {x}={ 0.21} 
Vertex at  {x,y} = { 0.21,-0.21} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.15, 0.00} 
Root 2 at  {x,y} = { 0.27, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.4     Solving   50x²-21x+2 = 0 by Completing The Square .

 Divide both sides of the equation by  50  to have 1 as the coefficient of the first term :
   x²-(21/50)x+(1/25) = 0

Subtract  1/25  from both side of the equation :
   x²-(21/50)x = -1/25

Now the clever bit: Take the coefficient of  x , which is  21/50 , divide by two, giving  21/100 , and finally square it giving  441/10000 

Add  441/10000  to both sides of the equation :
  On the right hand side we have :
   -1/25  +  441/10000   The common denominator of the two fractions is  10000   Adding  (-400/10000)+(441/10000)  gives  41/10000 
  So adding to both sides we finally get :
   x²-(21/50)x+(441/10000) = 41/10000

Adding  441/10000  has completed the left hand side into a perfect square :
   x²-(21/50)x+(441/10000)  =
   (x-(21/100)) • (x-(21/100))  =
  (x-(21/100))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(21/50)x+(441/10000) = 41/10000 and
   x²-(21/50)x+(441/10000) = (x-(21/100))²
then, according to the law of transitivity,
   (x-(21/100))² = 41/10000

We'll refer to this Equation as  Eq. #8.4.1  

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

Note that the square root of
   (x-(21/100))²   is
   (x-(21/100))^2/2 =
  (x-(21/100))¹ =
   x-(21/100)

Now, applying the Square Root Principle to  Eq. #8.4.1  we get:
   x-(21/100) = √ 41/10000

Add  21/100  to both sides to obtain:
   x = 21/100 + √ 41/10000

Since a square root has two values, one positive and the other negative
   x² - (21/50)x + (1/25) = 0
   has two solutions:
  x = 21/100 + √ 41/10000
   or
  x = 21/100 - √ 41/10000

Note that  √ 41/10000 can be written as
  √ 41  / √ 10000   which is √ 41  / 100

Solve Quadratic Equation using the Quadratic Formula

 8.5     Solving    50x²-21x+2 = 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   =     50
                      B   =   -21
                      C   =   2

Accordingly,  B²  -  4AC   =
                     441 - 400 =
                     41

Applying the quadratic formula :

               21 ± √ 41
   x  =    —————
                    100

  √ 41   , rounded to 4 decimal digits, is   6.4031
 So now we are looking at:
           x  =  ( 21 ±  6.403 ) / 100

Two real solutions:

 x =(21+√41)/100= 0.274

or:

 x =(21-√41)/100= 0.146

Two solutions were found :

1.  x =(21-√41)/100= 0.146
2.  x =(21+√41)/100= 0.274