Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "27.04" was replaced by "(2704/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            676
 Simplify   ———
            25 

Equation at the end of step  1  :

    611        35996     676
  ((———•(x2))+(—————•x))-———  = 0 
    100        1000      25 

Step  2  :

            8999
 Simplify   ————
            250 

Equation at the end of step  2  :

    611        8999     676
  ((———•(x2))+(————•x))-———  = 0 
    100        250      25 
 Step  3  :
            611
 Simplify   ———
            100

Equation at the end of step  3  :

    611          8999x     676
  ((——— • x2) +  —————) -  ———  = 0 
    100           250      25 

Step  4  :

Equation at the end of step  4  :

   611x2    8999x     676
  (————— +  —————) -  ———  = 0 
    100      250      25 

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       100 

      The right denominator is :       250 

      Least Common Multiple:
      500 

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 = 5

   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.      611x2 • 5
   ——————————————————  =   —————————
         L.C.M                500   

   R. Mult. • R. Num.      8999x • 2
   ——————————————————  =   —————————
         L.C.M                500   

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:

 611x2 • 5 + 8999x • 2     3055x2 + 17998x
 —————————————————————  =  ———————————————
          500                    500      

Equation at the end of step  5  :

  (3055x2 + 17998x)    676
  ————————————————— -  ———  = 0 
         500           25 

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   3055x² + 17998x  =   x • (3055x + 17998) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       500 

      The right denominator is :       25 

      Least Common Multiple:
      500 

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 = 20

Making Equivalent Fractions :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      x • (3055x+17998)
   ——————————————————  =   —————————————————
         L.C.M                    500       

   R. Mult. • R. Num.      676 • 20
   ——————————————————  =   ————————
         L.C.M               500   

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 x • (3055x+17998) - (676 • 20)     3055x2 + 17998x - 13520
 ——————————————————————————————  =  ———————————————————————
              500                             500          

Trying to factor by splitting the middle term

 7.6     Factoring  3055x² + 17998x - 13520 

The first term is,  3055x²  its coefficient is  3055 .
The middle term is,  +17998x  its coefficient is  17998 .
The last term, "the constant", is  -13520 

Step-1 : Multiply the coefficient of the first term by the constant   3055 • -13520 = -41303600 

Step-2 : Find two factors of  -41303600  whose sum equals the coefficient of the middle term, which is   17998 .


Numbers too big. Method shall not be applied

Equation at the end of step  7  :

  3055x2 + 17998x - 13520
  ———————————————————————  = 0 
            500          

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:

  3055x2+17998x-13520
  ——————————————————— • 500 = 0 • 500
          500        

Now, on the left hand side, the  500  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   3055x²+17998x-13520  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = 3055x²+17998x-13520

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, 3055 , 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  -2.9457  

 Plugging into the parabola formula  -2.9457  for  x  we can calculate the  y -coordinate : 
  y = 3055.0 * -2.95 * -2.95 + 17998.0 * -2.95 - 13520.0
or   y = -40028.020

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 3055x²+17998x-13520
Axis of Symmetry (dashed)  {x}={-2.95} 
Vertex at  {x,y} = {-2.95,-40028.02} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-6.57, 0.00} 
Root 2 at  {x,y} = { 0.67, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 8.3     Solving    3055x²+17998x-13520 = 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   =     3055.00
     B   =    17998.00
     C   =    -13520.00

   B² = 323928004.00 
   4AC = -165214400.00 
   B² - 4AC = 489142404.00 
   SQRT(B²-4AC) =  22116.56
   x=( -17998.00 ± 22116.56) / 6110.00 
   x =  0.67407
   x =  -6.56540

Two solutions were found :

1.    x =  -6.56540
   
2.    x =  0.67407