Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((25•52x2) -  309x) +  196  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  800x²-309x+196 

The first term is,  800x²  its coefficient is  800 .
The middle term is,  -309x  its coefficient is  -309 .
The last term, "the constant", is  +196 

Step-1 : Multiply the coefficient of the first term by the constant   800 • 196 = 156800 

Step-2 : Find two factors of  156800  whose sum equals the coefficient of the middle term, which is   -309 .

For tidiness, printing of 138 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  :

  800x2 - 309x + 196  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 800x²-309x+196

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, 800 , 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   0.1931  

 Plugging into the parabola formula   0.1931  for  x  we can calculate the  y -coordinate : 
  y = 800.0 * 0.19 * 0.19 - 309.0 * 0.19 + 196.0
or   y = 166.162

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 800x²-309x+196
Axis of Symmetry (dashed)  {x}={ 0.19} 
Vertex at  {x,y} = { 0.19,166.16} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 3.2     Solving   800x²-309x+196 = 0 by Completing The Square .

 Divide both sides of the equation by  800  to have 1 as the coefficient of the first term :
   x²-(309/800)x+(49/200) = 0

Subtract  49/200  from both side of the equation :
   x²-(309/800)x = -49/200

Now the clever bit: Take the coefficient of  x , which is  309/800 , divide by two, giving  309/1600 , and finally square it giving  309/1600 

Add  309/1600  to both sides of the equation :
  On the right hand side we have :
   -49/200  +  309/1600   The common denominator of the two fractions is  1600   Adding  (-392/1600)+(309/1600)  gives  -83/1600 
  So adding to both sides we finally get :
   x²-(309/800)x+(309/1600) = -83/1600

Adding  309/1600  has completed the left hand side into a perfect square :
   x²-(309/800)x+(309/1600)  =
   (x-(309/1600)) • (x-(309/1600))  =
  (x-(309/1600))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(309/800)x+(309/1600) = -83/1600 and
   x²-(309/800)x+(309/1600) = (x-(309/1600))²
then, according to the law of transitivity,
   (x-(309/1600))² = -83/1600

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-(309/1600))²   is
   (x-(309/1600))^2/2 =
  (x-(309/1600))¹ =
   x-(309/1600)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(309/1600) = √ -83/1600

Add  309/1600  to both sides to obtain:
   x = 309/1600 + √ -83/1600
In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 


Since a square root has two values, one positive and the other negative
   x² - (309/800)x + (49/200) = 0
   has two solutions:
  x = 309/1600 + √ 83/1600 •  i 
   or
  x = 309/1600 - √ 83/1600 •  i 

Note that  √ 83/1600 can be written as
  √ 83  / √ 1600   which is √ 83  / 40

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    800x²-309x+196 = 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   =    800
                      B   =   -309
                      C   =  196

Accordingly,  B²  -  4AC   =
                     95481 - 627200 =
                     -531719

Applying the quadratic formula :

               309 ± √ -531719
   x  =    —————————
                          1600

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -531719  = 
                    √ 531719 • (-1)  =
                    √ 531719  • √ -1   =
                    ±  √ 531719  • i

  √ 531719   , rounded to 4 decimal digits, is  729.1906
 So now we are looking at:
           x  =  ( 309 ±  729.191 i ) / 1600

Two imaginary solutions :

 x =(309+√-531719)/1600=(309+i√ 531719 )/1600= 0.1931+0.4557i
  or: 
 x =(309-√-531719)/1600=(309-i√ 531719 )/1600= 0.1931-0.4557i

Two solutions were found :

1.  x =(309-√-531719)/1600=(309-i√ 531719 )/1600= 0.1931-0.4557i
2.  x =(309+√-531719)/1600=(309+i√ 531719 )/1600= 0.1931+0.4557i