Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "1.001" was replaced by "(1001/1000)".

Step by step solution :

Step  1  :

            1001
 Simplify   ————
            1000

Equation at the end of step  1  :

                 1001
  ((x2) +  x) -  ————  = 0 
                 1000

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  1000  as the denominator :

               x2 + x     (x2 + x) • 1000
     x2 + x =  ——————  =  ———————————————
                 1             1000      

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  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² + x  =   x • (x + 1) 

Adding fractions that have a common denominator :

 3.2       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 • (x+1) • 1000 - (1001)     1000x2 + 1000x - 1001
 —————————————————————————  =  —————————————————————
           1000                        1000         

Trying to factor by splitting the middle term

 3.3     Factoring  1000x² + 1000x - 1001 

The first term is,  1000x²  its coefficient is  1000 .
The middle term is,  +1000x  its coefficient is  1000 .
The last term, "the constant", is  -1001 

Step-1 : Multiply the coefficient of the first term by the constant   1000 • -1001 = -1001000 

Step-2 : Find two factors of  -1001000  whose sum equals the coefficient of the middle term, which is   1000 .


Numbers too big. Method shall not be applied

Equation at the end of step  3  :

  1000x2 + 1000x - 1001
  —————————————————————  = 0 
          1000         

Step  4  :

When a fraction equals zero :

 4.1    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:

  1000x2+1000x-1001
  ————————————————— • 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 :
   1000x²+1000x-1001  = 0

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 1000x²+1000x-1001

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, 1000 , 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  -0.5000  

 Plugging into the parabola formula  -0.5000  for  x  we can calculate the  y -coordinate : 
  y = 1000.0 * -0.50 * -0.50 + 1000.0 * -0.50 - 1001.0
or   y = -1251.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 1000x²+1000x-1001
Axis of Symmetry (dashed)  {x}={-0.50} 
Vertex at  {x,y} = {-0.50,-1251.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.62, 0.00} 
Root 2 at  {x,y} = { 0.62, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    1000x²+1000x-1001 = 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   =     1000.00
     B   =    1000.00
     C   =    -1001.00

   B² = 1000000.00 
   4AC = -4004000.00 
   B² - 4AC = 5004000.00 
   SQRT(B²-4AC) =  2236.96
   x=( -1000.00 ± 2236.96) / 2000.00 
   x =  0.61848
   x =  -1.61848

Two solutions were found :

1.    x =  -1.61848
   
2.    x =  0.61848