Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "7.37" was replaced by "(737/100)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            737
 Simplify   ———
            100

Equation at the end of step  1  :

    362                   737
  ((——— • (y2)) -  2y) -  ———  = 0 
    100                   100
 Step  2  :
            181
 Simplify   ———
            50 

Equation at the end of step  2  :

    181                 737
  ((——— • y2) -  2y) -  ———  = 0 
    50                  100

Step  3  :

Equation at the end of step  3  :

   181y2           737
  (————— -  2y) -  ———  = 0 
    50             100

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  50  as the denominator :

          2y     2y • 50
    2y =  ——  =  ———————
          1        50   

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:

 181y2 - (2y • 50)     181y2 - 100y
 —————————————————  =  ————————————
        50                  50     

Equation at the end of step  4  :

  (181y2 - 100y)    737
  —————————————— -  ———  = 0 
        50          100

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   181y² - 100y  =   y • (181y - 100) 

Calculating the Least Common Multiple :

 6.2    Find the Least Common Multiple

      The left denominator is :       50 

      The right denominator is :       100 

      Least Common Multiple:
      100 

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 = 2

   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.      y • (181y-100) • 2
   ——————————————————  =   ——————————————————
         L.C.M                    100        

   R. Mult. • R. Num.      737
   ——————————————————  =   ———
         L.C.M             100

Adding fractions that have a common denominator :

 6.5       Adding up the two equivalent fractions

 y • (181y-100) • 2 - (737)     362y2 - 200y - 737
 ——————————————————————————  =  ——————————————————
            100                        100        

Trying to factor by splitting the middle term

 6.6     Factoring  362y² - 200y - 737 

The first term is,  362y²  its coefficient is  362 .
The middle term is,  -200y  its coefficient is  -200 .
The last term, "the constant", is  -737 

Step-1 : Multiply the coefficient of the first term by the constant   362 • -737 = -266794 

Step-2 : Find two factors of  -266794  whose sum equals the coefficient of the middle term, which is   -200 .

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

Equation at the end of step  6  :

  362y2 - 200y - 737
  ——————————————————  = 0 
         100        

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:

  362y2-200y-737
  —————————————— • 100 = 0 • 100
       100      

Now, on the left hand side, the  100  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   362y²-200y-737  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   t = 362y²-200y-737

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  "t"  because the coefficient of the first term, 362 , 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,Ay²+By+C,the  y -coordinate of the vertex is given by  -B/(2A) . In our case the  y  coordinate is   0.2762  

 Plugging into the parabola formula   0.2762  for  y  we can calculate the  t -coordinate : 
  t = 362.0 * 0.28 * 0.28 - 200.0 * 0.28 - 737.0
or   t = -764.624

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  t = 362y²-200y-737
Axis of Symmetry (dashed)  {y}={ 0.28} 
Vertex at  {y,t} = { 0.28,-764.62} 
 y -Intercepts (Roots) :
Root 1 at  {y,t} = {-1.18, 0.00} 
Root 2 at  {y,t} = { 1.73, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   362y²-200y-737 = 0 by Completing The Square .

 Divide both sides of the equation by  362  to have 1 as the coefficient of the first term :
   y²-(100/181)y-(737/362) = 0

Add  737/362  to both side of the equation :
   y²-(100/181)y = 737/362

Now the clever bit: Take the coefficient of  y , which is  100/181 , divide by two, giving  50/181 , and finally square it giving  2500/32761 

Add  2500/32761  to both sides of the equation :
  On the right hand side we have :
   737/362  +  2500/32761   The common denominator of the two fractions is  65522   Adding  (133397/65522)+(5000/65522)  gives  138397/65522 
  So adding to both sides we finally get :
   y²-(100/181)y+(2500/32761) = 138397/65522

Adding  2500/32761  has completed the left hand side into a perfect square :
   y²-(100/181)y+(2500/32761)  =
   (y-(50/181)) • (y-(50/181))  =
  (y-(50/181))²
Things which are equal to the same thing are also equal to one another. Since
   y²-(100/181)y+(2500/32761) = 138397/65522 and
   y²-(100/181)y+(2500/32761) = (y-(50/181))²
then, according to the law of transitivity,
   (y-(50/181))² = 138397/65522

We'll refer to this Equation as  Eq. #7.3.1  

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

Note that the square root of
   (y-(50/181))²   is
   (y-(50/181))^2/2 =
  (y-(50/181))¹ =
   y-(50/181)

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   y-(50/181) = √ 138397/65522

Add  50/181  to both sides to obtain:
   y = 50/181 + √ 138397/65522

Since a square root has two values, one positive and the other negative
   y² - (100/181)y - (737/362) = 0
   has two solutions:
  y = 50/181 + √ 138397/65522
   or
  y = 50/181 - √ 138397/65522

Note that  √ 138397/65522 can be written as
  √ 138397  / √ 65522 

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    362y²-200y-737 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  y  , the solution for   Ay²+By+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  y =   ————————
                      2A

  In our case,  A   =    362
                      B   =   -200
                      C   =  -737

Accordingly,  B²  -  4AC   =
                     40000 - (-1067176) =
                     1107176

Applying the quadratic formula :

               200 ± √ 1107176
   y  =    —————————
                          724

Can  √ 1107176 be simplified ?

Yes!   The prime factorization of  1107176   is
   2•2•2•7•17•1163 
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).

√ 1107176   =  √ 2•2•2•7•17•1163   =
                ±  2 • √ 276794

  √ 276794   , rounded to 4 decimal digits, is  526.1122
 So now we are looking at:
           y  =  ( 200 ± 2 •  526.112 ) / 724

Two real solutions:

 y =(200+√1107176)/724=50/181+1/362√ 276794 = 1.730

or:

 y =(200-√1107176)/724=50/181-1/362√ 276794 = -1.177

Two solutions were found :

1.  y =(200-√1107176)/724=50/181-1/362√ 276794 = -1.177
2.  y =(200+√1107176)/724=50/181+1/362√ 276794 = 1.730