Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "31.8" was replaced by "(318/10)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            159
 Simplify   ———
             5 

Equation at the end of step  1  :

    4905        225     159
  ((————•(x2))-(———•x))-———  = 0 
    1000        100      5 

Step  2  :

            9
 Simplify   —
            4

Equation at the end of step  2  :

    4905        9     159
  ((————•(x2))-(—•x))-———  = 0 
    1000        4      5 
 Step  3  :
            981
 Simplify   ———
            200

Equation at the end of step  3  :

    981          9x     159
  ((——— • x2) -  ——) -  ———  = 0 
    200          4       5 

Step  4  :

Equation at the end of step  4  :

   981x2    9x     159
  (————— -  ——) -  ———  = 0 
    200     4       5 

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       200 

      The right denominator is :       4 

      Least Common Multiple:
      200 

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 = 1

   Right_M = L.C.M / R_Deno = 50

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.      981x2
   ——————————————————  =   —————
         L.C.M              200 

   R. Mult. • R. Num.      9x • 50
   ——————————————————  =   ———————
         L.C.M               200  

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:

 981x2 - (9x • 50)     981x2 - 450x
 —————————————————  =  ————————————
        200                200     

Equation at the end of step  5  :

  (981x2 - 450x)    159
  —————————————— -  ———  = 0 
       200           5 

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   981x² - 450x  =   9x • (109x - 50) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       200 

      The right denominator is :       5 

      Least Common Multiple:
      200 

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 = 40

Making Equivalent Fractions :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      9x • (109x-50)
   ——————————————————  =   ——————————————
         L.C.M                  200      

   R. Mult. • R. Num.      159 • 40
   ——————————————————  =   ————————
         L.C.M               200   

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 9x • (109x-50) - (159 • 40)     981x2 - 450x - 6360
 ———————————————————————————  =  ———————————————————
             200                         200        

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   981x² - 450x - 6360  =   3 • (327x² - 150x - 2120) 

Trying to factor by splitting the middle term

 8.2     Factoring  327x² - 150x - 2120 

The first term is,  327x²  its coefficient is  327 .
The middle term is,  -150x  its coefficient is  -150 .
The last term, "the constant", is  -2120 

Step-1 : Multiply the coefficient of the first term by the constant   327 • -2120 = -693240 

Step-2 : Find two factors of  -693240  whose sum equals the coefficient of the middle term, which is   -150 .

For tidiness, printing of 58 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  8  :

  3 • (327x2 - 150x - 2120)
  —————————————————————————  = 0 
             200           

Step  9  :

When a fraction equals zero :

 9.1    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:

  3•(327x2-150x-2120)
  ——————————————————— • 200 = 0 • 200
          200        

Now, on the left hand side, the  200  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   3  •  (327x²-150x-2120)  = 0

Equations which are never true :

 9.2      Solve :    3   =  0

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

Parabola, Finding the Vertex :

 9.3      Find the Vertex of   y = 327x²-150x-2120

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, 327 , 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   0.2294  

 Plugging into the parabola formula   0.2294  for  x  we can calculate the  y -coordinate : 
  y = 327.0 * 0.23 * 0.23 - 150.0 * 0.23 - 2120.0
or   y = -2137.202

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 327x²-150x-2120
Axis of Symmetry (dashed)  {x}={ 0.23} 
Vertex at  {x,y} = { 0.23,-2137.20} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-2.33, 0.00} 
Root 2 at  {x,y} = { 2.79, 0.00} 

Solve Quadratic Equation by Completing The Square

 9.4     Solving   327x²-150x-2120 = 0 by Completing The Square .

 Divide both sides of the equation by  327  to have 1 as the coefficient of the first term :
   x²-(50/109)x-(2120/327) = 0

Add  2120/327  to both side of the equation :
   x²-(50/109)x = 2120/327

Now the clever bit: Take the coefficient of  x , which is  50/109 , divide by two, giving  25/109 , and finally square it giving  625/11881 

Add  625/11881  to both sides of the equation :
  On the right hand side we have :
   2120/327  +  625/11881   The common denominator of the two fractions is  35643   Adding  (231080/35643)+(1875/35643)  gives  232955/35643 
  So adding to both sides we finally get :
   x²-(50/109)x+(625/11881) = 232955/35643

Adding  625/11881  has completed the left hand side into a perfect square :
   x²-(50/109)x+(625/11881)  =
   (x-(25/109)) • (x-(25/109))  =
  (x-(25/109))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(50/109)x+(625/11881) = 232955/35643 and
   x²-(50/109)x+(625/11881) = (x-(25/109))²
then, according to the law of transitivity,
   (x-(25/109))² = 232955/35643

We'll refer to this Equation as  Eq. #9.4.1  

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

Note that the square root of
   (x-(25/109))²   is
   (x-(25/109))^2/2 =
  (x-(25/109))¹ =
   x-(25/109)

Now, applying the Square Root Principle to  Eq. #9.4.1  we get:
   x-(25/109) = √ 232955/35643

Add  25/109  to both sides to obtain:
   x = 25/109 + √ 232955/35643

Since a square root has two values, one positive and the other negative
   x² - (50/109)x - (2120/327) = 0
   has two solutions:
  x = 25/109 + √ 232955/35643
   or
  x = 25/109 - √ 232955/35643

Note that  √ 232955/35643 can be written as
  √ 232955  / √ 35643 

Solve Quadratic Equation using the Quadratic Formula

 9.5     Solving    327x²-150x-2120 = 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   =    327
                      B   =   -150
                      C   =  -2120

Accordingly,  B²  -  4AC   =
                     22500 - (-2772960) =
                     2795460

Applying the quadratic formula :

               150 ± √ 2795460
   x  =    —————————
                          654

Can  √ 2795460 be simplified ?

Yes!   The prime factorization of  2795460   is
   2•2•3•5•46591 
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).

√ 2795460   =  √ 2•2•3•5•46591   =
                ±  2 • √ 698865

  √ 698865   , rounded to 4 decimal digits, is  835.9815
 So now we are looking at:
           x  =  ( 150 ± 2 •  835.981 ) / 654

Two real solutions:

 x =(150+√2795460)/654=25/109+1/327√ 698865 = 2.786

or:

 x =(150-√2795460)/654=25/109-1/327√ 698865 = -2.327

Two solutions were found :

1.  x =(150-√2795460)/654=25/109-1/327√ 698865 = -2.327
2.  x =(150+√2795460)/654=25/109+1/327√ 698865 = 2.786