Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((22•53x2) +  99x) -  104  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  500x²+99x-104 

The first term is,  500x²  its coefficient is  500 .
The middle term is,  +99x  its coefficient is  99 .
The last term, "the constant", is  -104 

Step-1 : Multiply the coefficient of the first term by the constant   500 • -104 = -52000 

Step-2 : Find two factors of  -52000  whose sum equals the coefficient of the middle term, which is   99 .

For tidiness, printing of 42 lines which failed to find two such factors, was suppressed

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

Equation at the end of step  2  :

  500x2 + 99x - 104  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 500x²+99x-104

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, 500 , 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  -0.0990  

 Plugging into the parabola formula  -0.0990  for  x  we can calculate the  y -coordinate : 
  y = 500.0 * -0.10 * -0.10 + 99.0 * -0.10 - 104.0
or   y = -108.900

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 500x²+99x-104
Axis of Symmetry (dashed)  {x}={-0.10} 
Vertex at  {x,y} = {-0.10,-108.90} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.57, 0.00} 
Root 2 at  {x,y} = { 0.37, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   500x²+99x-104 = 0 by Completing The Square .

 Divide both sides of the equation by  500  to have 1 as the coefficient of the first term :
   x²+(99/500)x-(26/125) = 0

Add  26/125  to both side of the equation :
   x²+(99/500)x = 26/125

Now the clever bit: Take the coefficient of  x , which is  99/500 , divide by two, giving  99/1000 , and finally square it giving  99/1000 

Add  99/1000  to both sides of the equation :
  On the right hand side we have :
   26/125  +  99/1000   The common denominator of the two fractions is  1000   Adding  (208/1000)+(99/1000)  gives  307/1000 
  So adding to both sides we finally get :
   x²+(99/500)x+(99/1000) = 307/1000

Adding  99/1000  has completed the left hand side into a perfect square :
   x²+(99/500)x+(99/1000)  =
   (x+(99/1000)) • (x+(99/1000))  =
  (x+(99/1000))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(99/500)x+(99/1000) = 307/1000 and
   x²+(99/500)x+(99/1000) = (x+(99/1000))²
then, according to the law of transitivity,
   (x+(99/1000))² = 307/1000

We'll refer to this Equation as  Eq. #3.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x+(99/1000))²   is
   (x+(99/1000))^2/2 =
  (x+(99/1000))¹ =
   x+(99/1000)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x+(99/1000) = √ 307/1000

Subtract  99/1000  from both sides to obtain:
   x = -99/1000 + √ 307/1000

Since a square root has two values, one positive and the other negative
   x² + (99/500)x - (26/125) = 0
   has two solutions:
  x = -99/1000 + √ 307/1000
   or
  x = -99/1000 - √ 307/1000

Note that  √ 307/1000 can be written as
  √ 307  / √ 1000 

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    500x²+99x-104 = 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   =    500
                      B   =    99
                      C   =  -104

Accordingly,  B²  -  4AC   =
                     9801 - (-208000) =
                     217801

Applying the quadratic formula :

               -99 ± √ 217801
   x  =    ————————
                        1000

  √ 217801   , rounded to 4 decimal digits, is  466.6915
 So now we are looking at:
           x  =  ( -99 ±  466.692 ) / 1000

Two real solutions:

 x =(-99+√217801)/1000= 0.368

or:

 x =(-99-√217801)/1000=-0.566

Two solutions were found :

1.  x =(-99-√217801)/1000=-0.566
2.  x =(-99+√217801)/1000= 0.368