Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "3.56" was replaced by "(356/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            89
 Simplify   ——
            25

Equation at the end of step  1  :

    476        1087     89
  ((———•(x2))-(————•x))+——  = 0 
    100        100      25

Step  2  :

            1087
 Simplify   ————
            100 

Equation at the end of step  2  :

    476        1087     89
  ((———•(x2))-(————•x))+——  = 0 
    100        100      25
 Step  3  :
            119
 Simplify   ———
            25 

Equation at the end of step  3  :

    119          1087x     89
  ((——— • x2) -  —————) +  ——  = 0 
    25            100      25

Step  4  :

Equation at the end of step  4  :

   119x2    1087x     89
  (————— -  —————) +  ——  = 0 
    25       100      25

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       25 

      The right denominator is :       100 

      Least Common Multiple:
      100 

Calculating Multipliers :

 5.2    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 = 4

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 5.3      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.      119x2 • 4
   ——————————————————  =   —————————
         L.C.M                100   

   R. Mult. • R. Num.      1087x
   ——————————————————  =   —————
         L.C.M              100 

Adding fractions that have a common denominator :

 5.4       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:

 119x2 • 4 - (1087x)     476x2 - 1087x
 ———————————————————  =  —————————————
         100                  100     

Equation at the end of step  5  :

  (476x2 - 1087x)    89
  ——————————————— +  ——  = 0 
        100          25

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   476x² - 1087x  =   x • (476x - 1087) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       100 

      The right denominator is :       25 

      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 = 1

   Right_M = L.C.M / R_Deno = 4

Making Equivalent Fractions :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      x • (476x-1087)
   ——————————————————  =   ———————————————
         L.C.M                   100      

   R. Mult. • R. Num.      89 • 4
   ——————————————————  =   ——————
         L.C.M              100  

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 x • (476x-1087) + 89 • 4     476x2 - 1087x + 356
 ————————————————————————  =  ———————————————————
           100                        100        

Trying to factor by splitting the middle term

 7.6     Factoring  476x² - 1087x + 356 

The first term is,  476x²  its coefficient is  476 .
The middle term is,  -1087x  its coefficient is  -1087 .
The last term, "the constant", is  +356 

Step-1 : Multiply the coefficient of the first term by the constant   476 • 356 = 169456 

Step-2 : Find two factors of  169456  whose sum equals the coefficient of the middle term, which is   -1087 .

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  :

  476x2 - 1087x + 356
  ———————————————————  = 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:

  476x2-1087x+356
  ——————————————— • 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 :
   476x²-1087x+356  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = 476x²-1087x+356

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, 476 , 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   1.1418  

 Plugging into the parabola formula   1.1418  for  x  we can calculate the  y -coordinate : 
  y = 476.0 * 1.14 * 1.14 - 1087.0 * 1.14 + 356.0
or   y = -264.572

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 476x²-1087x+356
Axis of Symmetry (dashed)  {x}={ 1.14} 
Vertex at  {x,y} = { 1.14,-264.57} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.40, 0.00} 
Root 2 at  {x,y} = { 1.89, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   476x²-1087x+356 = 0 by Completing The Square .

 Divide both sides of the equation by  476  to have 1 as the coefficient of the first term :
   x²-(1087/476)x+(89/119) = 0

Subtract  89/119  from both side of the equation :
   x²-(1087/476)x = -89/119

Now the clever bit: Take the coefficient of  x , which is  1087/476 , divide by two, giving  1087/952 , and finally square it giving  1087/952 

Add  1087/952  to both sides of the equation :
  On the right hand side we have :
   -89/119  +  1087/952   The common denominator of the two fractions is  952   Adding  (-712/952)+(1087/952)  gives  375/952 
  So adding to both sides we finally get :
   x²-(1087/476)x+(1087/952) = 375/952

Adding  1087/952  has completed the left hand side into a perfect square :
   x²-(1087/476)x+(1087/952)  =
   (x-(1087/952)) • (x-(1087/952))  =
  (x-(1087/952))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(1087/476)x+(1087/952) = 375/952 and
   x²-(1087/476)x+(1087/952) = (x-(1087/952))²
then, according to the law of transitivity,
   (x-(1087/952))² = 375/952

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-(1087/952))²   is
   (x-(1087/952))^2/2 =
  (x-(1087/952))¹ =
   x-(1087/952)

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-(1087/952) = √ 375/952

Add  1087/952  to both sides to obtain:
   x = 1087/952 + √ 375/952

Since a square root has two values, one positive and the other negative
   x² - (1087/476)x + (89/119) = 0
   has two solutions:
  x = 1087/952 + √ 375/952
   or
  x = 1087/952 - √ 375/952

Note that  √ 375/952 can be written as
  √ 375  / √ 952 

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    476x²-1087x+356 = 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   =    476
                      B   =   -1087
                      C   =  356

Accordingly,  B²  -  4AC   =
                     1181569 - 677824 =
                     503745

Applying the quadratic formula :

               1087 ± √ 503745
   x  =    —————————
                          952

  √ 503745   , rounded to 4 decimal digits, is  709.7500
 So now we are looking at:
           x  =  ( 1087 ±  709.750 ) / 952

Two real solutions:

 x =(1087+√503745)/952= 1.887

or:

 x =(1087-√503745)/952= 0.396

Two solutions were found :

1.  x =(1087-√503745)/952= 0.396
2.  x =(1087+√503745)/952= 1.887