Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "27.3" was replaced by "(273/10)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            273
 Simplify   ———
            10 

Equation at the end of step  1  :

       981         273
  ((0-(———•(x2)))+(———•x))+1  = 0 
       100         10 
 Step  2  :
            981
 Simplify   ———
            100

Equation at the end of step  2  :

          981           273x     
  ((0 -  (——— • x2)) +  ————) +  1  = 0 
          100            10      

Step  3  :

Equation at the end of step  3  :

         981x2     273x     
  ((0 -  —————) +  ————) +  1  = 0 
          100       10      

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       100 

      The right denominator is :       10 

      Least Common Multiple:
      100 

Calculating Multipliers :

 4.2    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 = 10

Making Equivalent Fractions :

 4.3      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              100  

   R. Mult. • R. Num.      273x • 10
   ——————————————————  =   —————————
         L.C.M                100   

Adding fractions that have a common denominator :

 4.4       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 + 273x • 10     2730x - 981x2
 ——————————————————  =  —————————————
        100                  100     

Equation at the end of step  4  :

  (2730x - 981x2)    
  ——————————————— +  1  = 0 
        100          

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  100  as the denominator :

         1     1 • 100
    1 =  —  =  ———————
         1       100  

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

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   2730x - 981x²  =   -3x • (327x - 910) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -3x • (327x-910) + 100     -981x2 + 2730x + 100
 ——————————————————————  =  ————————————————————
          100                       100         

Trying to factor by splitting the middle term

 6.3     Factoring  -981x² + 2730x + 100 

The first term is,  -981x²  its coefficient is  -981 .
The middle term is,  +2730x  its coefficient is  2730 .
The last term, "the constant", is  +100 

Step-1 : Multiply the coefficient of the first term by the constant   -981 • 100 = -98100 

Step-2 : Find two factors of  -98100  whose sum equals the coefficient of the middle term, which is   2730 .

For tidiness, printing of 48 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  6  :

  -981x2 + 2730x + 100
  ————————————————————  = 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:

  -981x2+2730x+100
  ———————————————— • 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 :
   -981x²+2730x+100  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = -981x²+2730x+100

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, -981 , 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   1.3914  

 Plugging into the parabola formula   1.3914  for  x  we can calculate the  y -coordinate : 
  y = -981.0 * 1.39 * 1.39 + 2730.0 * 1.39 + 100.0
or   y = 1999.312

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -981x²+2730x+100
Axis of Symmetry (dashed)  {x}={ 1.39} 
Vertex at  {x,y} = { 1.39,1999.31} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 2.82, 0.00} 
Root 2 at  {x,y} = {-0.04, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   -981x²+2730x+100 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 981x²-2730x-100 = 0  Divide both sides of the equation by  981  to have 1 as the coefficient of the first term :
   x²-(910/327)x-(100/981) = 0

Add  100/981  to both side of the equation :
   x²-(910/327)x = 100/981

Now the clever bit: Take the coefficient of  x , which is  910/327 , divide by two, giving  455/327 , and finally square it giving  207025/106929 

Add  207025/106929  to both sides of the equation :
  On the right hand side we have :
   100/981  +  207025/106929   The common denominator of the two fractions is  106929   Adding  (10900/106929)+(207025/106929)  gives  217925/106929 
  So adding to both sides we finally get :
   x²-(910/327)x+(207025/106929) = 217925/106929

Adding  207025/106929  has completed the left hand side into a perfect square :
   x²-(910/327)x+(207025/106929)  =
   (x-(455/327)) • (x-(455/327))  =
  (x-(455/327))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(910/327)x+(207025/106929) = 217925/106929 and
   x²-(910/327)x+(207025/106929) = (x-(455/327))²
then, according to the law of transitivity,
   (x-(455/327))² = 217925/106929

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
   (x-(455/327))²   is
   (x-(455/327))^2/2 =
  (x-(455/327))¹ =
   x-(455/327)

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   x-(455/327) = √ 217925/106929

Add  455/327  to both sides to obtain:
   x = 455/327 + √ 217925/106929

Since a square root has two values, one positive and the other negative
   x² - (910/327)x - (100/981) = 0
   has two solutions:
  x = 455/327 + √ 217925/106929
   or
  x = 455/327 - √ 217925/106929

Note that  √ 217925/106929 can be written as
  √ 217925  / √ 106929   which is √ 217925  / 327

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    -981x²+2730x+100 = 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   =    -981
                      B   =   2730
                      C   =  100

Accordingly,  B²  -  4AC   =
                     7452900 - (-392400) =
                     7845300

Applying the quadratic formula :

               -2730 ± √ 7845300
   x  =    ——————————
                           -1962

Can  √ 7845300 be simplified ?

Yes!   The prime factorization of  7845300   is
   2•2•3•3•5•5•23•379 
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).

√ 7845300   =  √ 2•2•3•3•5•5•23•379   =2•3•5•√ 8717   =
                ±  30 • √ 8717

  √ 8717   , rounded to 4 decimal digits, is  93.3649
 So now we are looking at:
           x  =  ( -2730 ± 30 •  93.365 ) / -1962

Two real solutions:

 x =(-2730+√7845300)/-1962=(455-5√ 8717 )/327= -0.036

or:

 x =(-2730-√7845300)/-1962=(455+5√ 8717 )/327= 2.819

Two solutions were found :

1.  x =(-2730-√7845300)/-1962=(455+5√ 8717 )/327= 2.819
2.  x =(-2730+√7845300)/-1962=(455-5√ 8717 )/327= -0.036