Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "5.33" was replaced by "(533/100)". 3 more similar replacement(s)

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

           (766/100)*x^2+(274/100)*x-((533/100))=0 

Step by step solution :

Step  1  :

            533
 Simplify   ———
            100

Equation at the end of step  1  :

    766        274     533
  ((———•(x2))+(———•x))-———  = 0 
    100        100     100

Step  2  :

            137
 Simplify   ———
            50 

Equation at the end of step  2  :

    766        137     533
  ((———•(x2))+(———•x))-———  = 0 
    100        50      100
 Step  3  :
            383
 Simplify   ———
            50 

Equation at the end of step  3  :

    383          137x     533
  ((——— • x2) +  ————) -  ———  = 0 
    50            50      100

Step  4  :

Equation at the end of step  4  :

   383x2    137x     533
  (————— +  ————) -  ———  = 0 
    50       50      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:

 383x2 + 137x     383x2 + 137x
 ————————————  =  ————————————
      50               50     

Equation at the end of step  5  :

  (383x2 + 137x)    533
  —————————————— -  ———  = 0 
        50          100

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   383x² + 137x  =   x • (383x + 137) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       50 

      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 = 2

   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 • (383x+137) • 2
   ——————————————————  =   ——————————————————
         L.C.M                    100        

   R. Mult. • R. Num.      533
   ——————————————————  =   ———
         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 • (383x+137) • 2 - (533)     766x2 + 274x - 533
 ——————————————————————————  =  ——————————————————
            100                        100        

Trying to factor by splitting the middle term

 7.6     Factoring  766x² + 274x - 533 

The first term is,  766x²  its coefficient is  766 .
The middle term is,  +274x  its coefficient is  274 .
The last term, "the constant", is  -533 

Step-1 : Multiply the coefficient of the first term by the constant   766 • -533 = -408278 

Step-2 : Find two factors of  -408278  whose sum equals the coefficient of the middle term, which is   274 .

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

Equation at the end of step  7  :

  766x2 + 274x - 533
  ——————————————————  = 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:

  766x2+274x-533
  —————————————— • 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 :
   766x²+274x-533  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = 766x²+274x-533

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, 766 , 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  -0.1789  

 Plugging into the parabola formula  -0.1789  for  x  we can calculate the  y -coordinate : 
  y = 766.0 * -0.18 * -0.18 + 274.0 * -0.18 - 533.0
or   y = -557.503

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 766x²+274x-533
Axis of Symmetry (dashed)  {x}={-0.18} 
Vertex at  {x,y} = {-0.18,-557.50} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.03, 0.00} 
Root 2 at  {x,y} = { 0.67, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   766x²+274x-533 = 0 by Completing The Square .

 Divide both sides of the equation by  766  to have 1 as the coefficient of the first term :
   x²+(137/383)x-(533/766) = 0

Add  533/766  to both side of the equation :
   x²+(137/383)x = 533/766

Now the clever bit: Take the coefficient of  x , which is  137/383 , divide by two, giving  137/766 , and finally square it giving  137/766 

Add  137/766  to both sides of the equation :
  On the right hand side we have :
   533/766  +  137/766   The common denominator of the two fractions is  766   Adding  (533/766)+(137/766)  gives  670/766 
  So adding to both sides we finally get :
   x²+(137/383)x+(137/766) = 335/383

Adding  137/766  has completed the left hand side into a perfect square :
   x²+(137/383)x+(137/766)  =
   (x+(137/766)) • (x+(137/766))  =
  (x+(137/766))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(137/383)x+(137/766) = 335/383 and
   x²+(137/383)x+(137/766) = (x+(137/766))²
then, according to the law of transitivity,
   (x+(137/766))² = 335/383

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+(137/766))²   is
   (x+(137/766))^2/2 =
  (x+(137/766))¹ =
   x+(137/766)

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x+(137/766) = √ 335/383

Subtract  137/766  from both sides to obtain:
   x = -137/766 + √ 335/383

Since a square root has two values, one positive and the other negative
   x² + (137/383)x - (533/766) = 0
   has two solutions:
  x = -137/766 + √ 335/383
   or
  x = -137/766 - √ 335/383

Note that  √ 335/383 can be written as
  √ 335  / √ 383 

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    766x²+274x-533 = 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   =    766
                      B   =   274
                      C   =  -533

Accordingly,  B²  -  4AC   =
                     75076 - (-1633112) =
                     1708188

Applying the quadratic formula :

               -274 ± √ 1708188
   x  =    ——————————
                           1532

Can  √ 1708188 be simplified ?

Yes!   The prime factorization of  1708188   is
   2•2•3•283•503 
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).

√ 1708188   =  √ 2•2•3•283•503   =
                ±  2 • √ 427047

  √ 427047   , rounded to 4 decimal digits, is  653.4883
 So now we are looking at:
           x  =  ( -274 ± 2 •  653.488 ) / 1532

Two real solutions:

 x =(-274+√1708188)/1532=(-137+√ 427047 )/766= 0.674

or:

 x =(-274-√1708188)/1532=(-137-√ 427047 )/766= -1.032

Two solutions were found :

1.  x =(-274-√1708188)/1532=(-137-√ 427047 )/766= -1.032
2.  x =(-274+√1708188)/1532=(-137+√ 427047 )/766= 0.674