Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "12.37" was replaced by "(1237/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            1237
 Simplify   ————
            100 

Equation at the end of step  1  :

        5          5     1237
  ((0-(——•(x2)))+(——•x))+————  = 0 
       10         10     100 

Step  2  :

            1
 Simplify   —
            2

Equation at the end of step  2  :

        5         1     1237
  ((0-(——•(x2)))+(—•x))+————  = 0 
       10         2     100 
 Step  3  :
            1
 Simplify   —
            2

Equation at the end of step  3  :

          1           x     1237
  ((0 -  (— • x2)) +  —) +  ————  = 0 
          2           2     100 

Step  4  :

Equation at the end of step  4  :

         x2     x     1237
  ((0 -  ——) +  —) +  ————  = 0 
         2      2     100 

Step  5  :

Adding fractions which have a common denominator :

 5.1       Adding fractions which 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 + x     x - x2
 ———————  =  ——————
    2          2   

Equation at the end of step  5  :

  (x - x2)    1237
  ———————— +  ————  = 0 
     2        100 

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   x - x²  =   -x • (x - 1) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       100 

      Least Common Multiple:
      100 

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 = 50

   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 • (x-1) • 50
   ——————————————————  =   ———————————————
         L.C.M                   100      

   R. Mult. • R. Num.      1237
   ——————————————————  =   ————
         L.C.M             100 

Adding fractions that have a common denominator :

 7.5       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:

 -x • (x-1) • 50 + 1237     -50x2 + 50x + 1237
 ——————————————————————  =  ——————————————————
          100                      100        

Trying to factor by splitting the middle term

 7.6     Factoring  -50x² + 50x + 1237 

The first term is,  -50x²  its coefficient is  -50 .
The middle term is,  +50x  its coefficient is  50 .
The last term, "the constant", is  +1237 

Step-1 : Multiply the coefficient of the first term by the constant   -50 • 1237 = -61850 

Step-2 : Find two factors of  -61850  whose sum equals the coefficient of the middle term, which is   50 .

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

Equation at the end of step  7  :

  -50x2 + 50x + 1237
  ——————————————————  = 0 
         100        

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:

  -50x2+50x+1237
  —————————————— • 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 :
   -50x²+50x+1237  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -50x²+50x+1237

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, -50 , 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   0.5000  

 Plugging into the parabola formula   0.5000  for  x  we can calculate the  y -coordinate : 
  y = -50.0 * 0.50 * 0.50 + 50.0 * 0.50 + 1237.0
or   y = 1249.500

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -50x²+50x+1237
Axis of Symmetry (dashed)  {x}={ 0.50} 
Vertex at  {x,y} = { 0.50,1249.50} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 5.50, 0.00} 
Root 2 at  {x,y} = {-4.50, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   -50x²+50x+1237 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 50x²-50x-1237 = 0  Divide both sides of the equation by  50  to have 1 as the coefficient of the first term :
   x²-x-(1237/50) = 0

Add  1237/50  to both side of the equation :
   x²-x = 1237/50

Now the clever bit: Take the coefficient of  x , which is  1 , divide by two, giving  1/2 , and finally square it giving  1/4 

Add  1/4  to both sides of the equation :
  On the right hand side we have :
   1237/50  +  1/4   The common denominator of the two fractions is  100   Adding  (2474/100)+(25/100)  gives  2499/100 
  So adding to both sides we finally get :
   x²-x+(1/4) = 2499/100

Adding  1/4  has completed the left hand side into a perfect square :
   x²-x+(1/4)  =
   (x-(1/2)) • (x-(1/2))  =
  (x-(1/2))²
Things which are equal to the same thing are also equal to one another. Since
   x²-x+(1/4) = 2499/100 and
   x²-x+(1/4) = (x-(1/2))²
then, according to the law of transitivity,
   (x-(1/2))² = 2499/100

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

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-(1/2) = √ 2499/100

Add  1/2  to both sides to obtain:
   x = 1/2 + √ 2499/100

Since a square root has two values, one positive and the other negative
   x² - x - (1237/50) = 0
   has two solutions:
  x = 1/2 + √ 2499/100
   or
  x = 1/2 - √ 2499/100

Note that  √ 2499/100 can be written as
  √ 2499  / √ 100   which is √ 2499  / 10

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    -50x²+50x+1237 = 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   =    50
                      C   =  1237

Accordingly,  B²  -  4AC   =
                     2500 - (-247400) =
                     249900

Applying the quadratic formula :

               -50 ± √ 249900
   x  =    ————————
                        -100

Can  √ 249900 be simplified ?

Yes!   The prime factorization of  249900   is
   2•2•3•5•5•7•7•17 
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).

√ 249900   =  √ 2•2•3•5•5•7•7•17   =2•5•7•√ 51   =
                ±  70 • √ 51

  √ 51   , rounded to 4 decimal digits, is   7.1414
 So now we are looking at:
           x  =  ( -50 ± 70 •  7.141 ) / -100

Two real solutions:

 x =(-50+√249900)/-100=1/2-7/10√ 51 = -4.499

or:

 x =(-50-√249900)/-100=1/2+7/10√ 51 = 5.499

Two solutions were found :

1.  x =(-50-√249900)/-100=1/2+7/10√ 51 = 5.499
2.  x =(-50+√249900)/-100=1/2-7/10√ 51 = -4.499