Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "1.08" was replaced by "(108/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            27
 Simplify   ——
            25

Equation at the end of step  1  :

    409        314     27
  ((———•(x2))-(———•x))-——  = 0 
    100        100     25

Step  2  :

            157
 Simplify   ———
            50 

Equation at the end of step  2  :

    409        157     27
  ((———•(x2))-(———•x))-——  = 0 
    100        50      25
 Step  3  :
            409
 Simplify   ———
            100

Equation at the end of step  3  :

    409          157x     27
  ((——— • x2) -  ————) -  ——  = 0 
    100           50      25

Step  4  :

Equation at the end of step  4  :

   409x2    157x     27
  (————— -  ————) -  ——  = 0 
    100      50      25

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       100 

      The right denominator is :       50 

      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 = 1

   Right_M = L.C.M / R_Deno = 2

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.      409x2
   ——————————————————  =   —————
         L.C.M              100 

   R. Mult. • R. Num.      157x • 2
   ——————————————————  =   ————————
         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:

 409x2 - (157x • 2)     409x2 - 314x
 ——————————————————  =  ————————————
        100                 100     

Equation at the end of step  5  :

  (409x2 - 314x)    27
  —————————————— -  ——  = 0 
       100          25

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   409x² - 314x  =   x • (409x - 314) 

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 • (409x-314)
   ——————————————————  =   ——————————————
         L.C.M                  100      

   R. Mult. • R. Num.      27 • 4
   ——————————————————  =   ——————
         L.C.M              100  

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 x • (409x-314) - (27 • 4)     409x2 - 314x - 108
 —————————————————————————  =  ——————————————————
            100                       100        

Trying to factor by splitting the middle term

 7.6     Factoring  409x² - 314x - 108 

The first term is,  409x²  its coefficient is  409 .
The middle term is,  -314x  its coefficient is  -314 .
The last term, "the constant", is  -108 

Step-1 : Multiply the coefficient of the first term by the constant   409 • -108 = -44172 

Step-2 : Find two factors of  -44172  whose sum equals the coefficient of the middle term, which is   -314 .

For tidiness, printing of 18 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  :

  409x2 - 314x - 108
  ——————————————————  = 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:

  409x2-314x-108
  —————————————— • 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 :
   409x²-314x-108  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = 409x²-314x-108

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, 409 , 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   0.3839  

 Plugging into the parabola formula   0.3839  for  x  we can calculate the  y -coordinate : 
  y = 409.0 * 0.38 * 0.38 - 314.0 * 0.38 - 108.0
or   y = -168.267

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 409x²-314x-108
Axis of Symmetry (dashed)  {x}={ 0.38} 
Vertex at  {x,y} = { 0.38,-168.27} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.26, 0.00} 
Root 2 at  {x,y} = { 1.03, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   409x²-314x-108 = 0 by Completing The Square .

 Divide both sides of the equation by  409  to have 1 as the coefficient of the first term :
   x²-(314/409)x-(108/409) = 0

Add  108/409  to both side of the equation :
   x²-(314/409)x = 108/409

Now the clever bit: Take the coefficient of  x , which is  314/409 , divide by two, giving  157/409 , and finally square it giving  24649/167281 

Add  24649/167281  to both sides of the equation :
  On the right hand side we have :
   108/409  +  24649/167281   The common denominator of the two fractions is  167281   Adding  (44172/167281)+(24649/167281)  gives  68821/167281 
  So adding to both sides we finally get :
   x²-(314/409)x+(24649/167281) = 68821/167281

Adding  24649/167281  has completed the left hand side into a perfect square :
   x²-(314/409)x+(24649/167281)  =
   (x-(157/409)) • (x-(157/409))  =
  (x-(157/409))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(314/409)x+(24649/167281) = 68821/167281 and
   x²-(314/409)x+(24649/167281) = (x-(157/409))²
then, according to the law of transitivity,
   (x-(157/409))² = 68821/167281

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-(157/409))²   is
   (x-(157/409))^2/2 =
  (x-(157/409))¹ =
   x-(157/409)

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-(157/409) = √ 68821/167281

Add  157/409  to both sides to obtain:
   x = 157/409 + √ 68821/167281

Since a square root has two values, one positive and the other negative
   x² - (314/409)x - (108/409) = 0
   has two solutions:
  x = 157/409 + √ 68821/167281
   or
  x = 157/409 - √ 68821/167281

Note that  √ 68821/167281 can be written as
  √ 68821  / √ 167281   which is √ 68821  / 409

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    409x²-314x-108 = 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   =    409
                      B   =   -314
                      C   =  -108

Accordingly,  B²  -  4AC   =
                     98596 - (-176688) =
                     275284

Applying the quadratic formula :

               314 ± √ 275284
   x  =    ————————
                        818

Can  √ 275284 be simplified ?

Yes!   The prime factorization of  275284   is
   2•2•68821 
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).

√ 275284   =  √ 2•2•68821   =
                ±  2 • √ 68821

  √ 68821   , rounded to 4 decimal digits, is  262.3376
 So now we are looking at:
           x  =  ( 314 ± 2 •  262.338 ) / 818

Two real solutions:

 x =(314+√275284)/818=(157+√ 68821 )/409= 1.025

or:

 x =(314-√275284)/818=(157-√ 68821 )/409= -0.258

Two solutions were found :

1.  x =(314-√275284)/818=(157-√ 68821 )/409= -0.258
2.  x =(314+√275284)/818=(157+√ 68821 )/409= 1.025