Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "3.63" was replaced by "(363/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            363
 Simplify   ———
            100

Equation at the end of step  1  :

        931         7298     363
  ((0-(————•(x2)))+(————•x))+———  = 0 
       1000         1000     100

Step  2  :

            3649
 Simplify   ————
            500 

Equation at the end of step  2  :

        931         3649     363
  ((0-(————•(x2)))+(————•x))+———  = 0 
       1000         500      100
 Step  3  :
             931
 Simplify   ————
            1000

Equation at the end of step  3  :

           931           3649x     363
  ((0 -  (———— • x2)) +  —————) +  ———  = 0 
          1000            500      100

Step  4  :

Equation at the end of step  4  :

         931x2     3649x     363
  ((0 -  —————) +  —————) +  ———  = 0 
         1000       500      100

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       1000 

      The right denominator is :       500 

      Least Common Multiple:
      1000 

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.      -931x2
   ——————————————————  =   ——————
         L.C.M              1000 

   R. Mult. • R. Num.      3649x • 2
   ——————————————————  =   —————————
         L.C.M               1000   

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:

 -931x2 + 3649x • 2     7298x - 931x2
 ——————————————————  =  —————————————
        1000                1000     

Equation at the end of step  5  :

  (7298x - 931x2)    363
  ——————————————— +  ———  = 0 
       1000          100

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   7298x - 931x²  =   -x • (931x - 7298) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       1000 

      The right denominator is :       100 

      Least Common Multiple:
      1000 

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 = 10

Making Equivalent Fractions :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      -x • (931x-7298)
   ——————————————————  =   ————————————————
         L.C.M                   1000      

   R. Mult. • R. Num.      363 • 10
   ——————————————————  =   ————————
         L.C.M               1000  

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 -x • (931x-7298) + 363 • 10     -931x2 + 7298x + 3630
 ———————————————————————————  =  —————————————————————
            1000                         1000         

Trying to factor by splitting the middle term

 7.6     Factoring  -931x² + 7298x + 3630 

The first term is,  -931x²  its coefficient is  -931 .
The middle term is,  +7298x  its coefficient is  7298 .
The last term, "the constant", is  +3630 

Step-1 : Multiply the coefficient of the first term by the constant   -931 • 3630 = -3379530 

Step-2 : Find two factors of  -3379530  whose sum equals the coefficient of the middle term, which is   7298 .


Numbers too big. Method shall not be applied

Equation at the end of step  7  :

  -931x2 + 7298x + 3630
  —————————————————————  = 0 
          1000         

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:

  -931x2+7298x+3630
  ————————————————— • 1000 = 0 • 1000
        1000       

Now, on the left hand side, the  1000  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -931x²+7298x+3630  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = -931x²+7298x+3630

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, -931 , 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   3.9194  

 Plugging into the parabola formula   3.9194  for  x  we can calculate the  y -coordinate : 
  y = -931.0 * 3.92 * 3.92 + 7298.0 * 3.92 + 3630.0
or   y = 17932.042

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -931x²+7298x+3630
Axis of Symmetry (dashed)  {x}={ 3.92} 
Vertex at  {x,y} = { 3.92,17932.04} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 8.31, 0.00} 
Root 2 at  {x,y} = {-0.47, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 8.3     Solving    -931x²+7298x+3630 = 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   =     -931.00
     B   =    7298.00
     C   =    3630.00

   B² = 53260804.00 
   4AC = -13518120.00 
   B² - 4AC = 66778924.00 
   SQRT(B²-4AC) =  8171.84
   x=( -7298.00 ± 8171.84) / -1862.00 
   x =  -0.46930
   x =  8.30818

Two solutions were found :

1.    x =  8.30818
   
2.    x =  -0.46930