Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.001" was replaced by "(001/1000)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

              1 
 Simplify   ————
            1000

Equation at the end of step  1  :

      1                         1 
  ((———— • (x2)) +  1000x) +  ————  = 0 
    1000                      1000
 Step  2  :
              1 
 Simplify   ————
            1000

Equation at the end of step  2  :

      1                       1 
  ((———— • x2) +  1000x) +  ————  = 0 
    1000                    1000

Step  3  :

Equation at the end of step  3  :

    x2                 1 
  (———— +  1000x) +  ————  = 0 
   1000              1000

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

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

             1000x     1000x • 1000
    1000x =  —————  =  ————————————
               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

Adding fractions that have a common denominator :

 4.2       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:

 x2 + 1000x • 1000     x2 + 1000000x
 —————————————————  =  —————————————
       1000                1000     

Equation at the end of step  4  :

  (x2 + 1000000x)      1 
  ——————————————— +  ————  = 0 
       1000          1000

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   x² + 1000000x  =   x • (x + 1000000) 

Adding fractions which have a common denominator :

 6.2       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:

 x • (x+1000000) + 1     x2 + 1000000x + 1
 ———————————————————  =  —————————————————
        1000                   1000       

Trying to factor by splitting the middle term

 6.3     Factoring  x² + 1000000x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +1000000x  its coefficient is  1000000 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   1000000 .

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

Equation at the end of step  6  :

  x2 + 1000000x + 1
  —————————————————  = 0 
        1000       

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:

  x2+1000000x+1
  ————————————— • 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 :
   x²+1000000x+1  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = x²+1000000x+1

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, 1 , 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  -500000.0000  

 Plugging into the parabola formula  -500000.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * -500000.00 * -500000.00 + 1000000.0 * -500000.00 + 1.0
or   y = -249999999999.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x²+1000000x+1
Axis of Symmetry (dashed)  {x}={-500000.00} 
Vertex at  {x,y} = {-500000.00,-249999999999.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1000000.00, 0.00} 
Root 2 at  {x,y} = {-0.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   x²+1000000x+1 = 0 by Completing The Square .

 Subtract  1  from both side of the equation :
   x²+1000000x = -1

Now the clever bit: Take the coefficient of  x , which is  4618685/16777216 , divide by two, giving  4618685/16777216 , and finally square it giving  -851430007/0 

We are sorry, Numbers involved are too big to be handled by a little Tiger !!

We will be using another method to solve the equation

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    x²+1000000x+1 = 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   =     1
                      B   =   1000000
                      C   =   1

Accordingly,  B²  -  4AC   =
                     -727379968 - 4 =
                     -727379972

Applying the quadratic formula :

               -1000000 ± √ -727379972
   x  =    —————————————
                               2

In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 
Accordingly,√ -727379972  = 
                    √ 727379972 • (-1)  =
                    √ 727379972  • √ -1   =
                    ±  √ 727379972  • i

Can  √ 727379972 be simplified ?

Yes!   The prime factorization of  727379972   is
   2•2•11•29•570047 
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).

√ 727379972   =  √ 2•2•11•29•570047   =
                ±  2 • √ 181844993

  √ 181844993   , rounded to 4 decimal digits, is  13484.9914
 So now we are looking at:
           x  =  ( -1000000 ± 2 •  13484.991 i ) / 2

Two imaginary solutions :

 x =(-1000000+√-727379972)/2=-500000+i√ 181844993 = -500000.0000+13484.9914i
  or: 
 x =(-1000000-√-727379972)/2=-500000-i√ 181844993 = -500000.0000-13484.9914i

Two solutions were found :

1.  x =(-1000000-√-727379972)/2=-500000-i√ 181844993 = -500000.0000-13484.9914i
2.  x =(-1000000+√-727379972)/2=-500000+i√ 181844993 = -500000.0000+13484.9914i