Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•11x2) -  163x) -  210  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  22x²-163x-210 

The first term is,  22x²  its coefficient is  22 .
The middle term is,  -163x  its coefficient is  -163 .
The last term, "the constant", is  -210 

Step-1 : Multiply the coefficient of the first term by the constant   22 • -210 = -4620 

Step-2 : Find two factors of  -4620  whose sum equals the coefficient of the middle term, which is   -163 .

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  :

  22x2 - 163x - 210  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 22x²-163x-210

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, 22 , 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   3.7045  

 Plugging into the parabola formula   3.7045  for  x  we can calculate the  y -coordinate : 
  y = 22.0 * 3.70 * 3.70 - 163.0 * 3.70 - 210.0
or   y = -511.920

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 22x²-163x-210
Axis of Symmetry (dashed)  {x}={ 3.70} 
Vertex at  {x,y} = { 3.70,-511.92} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.12, 0.00} 
Root 2 at  {x,y} = { 8.53, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   22x²-163x-210 = 0 by Completing The Square .

 Divide both sides of the equation by  22  to have 1 as the coefficient of the first term :
   x²-(163/22)x-(105/11) = 0

Add  105/11  to both side of the equation :
   x²-(163/22)x = 105/11

Now the clever bit: Take the coefficient of  x , which is  163/22 , divide by two, giving  163/44 , and finally square it giving  26569/1936 

Add  26569/1936  to both sides of the equation :
  On the right hand side we have :
   105/11  +  26569/1936   The common denominator of the two fractions is  1936   Adding  (18480/1936)+(26569/1936)  gives  45049/1936 
  So adding to both sides we finally get :
   x²-(163/22)x+(26569/1936) = 45049/1936

Adding  26569/1936  has completed the left hand side into a perfect square :
   x²-(163/22)x+(26569/1936)  =
   (x-(163/44)) • (x-(163/44))  =
  (x-(163/44))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(163/22)x+(26569/1936) = 45049/1936 and
   x²-(163/22)x+(26569/1936) = (x-(163/44))²
then, according to the law of transitivity,
   (x-(163/44))² = 45049/1936

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-(163/44))²   is
   (x-(163/44))^2/2 =
  (x-(163/44))¹ =
   x-(163/44)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(163/44) = √ 45049/1936

Add  163/44  to both sides to obtain:
   x = 163/44 + √ 45049/1936

Since a square root has two values, one positive and the other negative
   x² - (163/22)x - (105/11) = 0
   has two solutions:
  x = 163/44 + √ 45049/1936
   or
  x = 163/44 - √ 45049/1936

Note that  √ 45049/1936 can be written as
  √ 45049  / √ 1936   which is √ 45049  / 44

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    22x²-163x-210 = 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   =     22
                      B   =   -163
                      C   =  -210

Accordingly,  B²  -  4AC   =
                     26569 - (-18480) =
                     45049

Applying the quadratic formula :

               163 ± √ 45049
   x  =    ————————
                        44

  √ 45049   , rounded to 4 decimal digits, is  212.2475
 So now we are looking at:
           x  =  ( 163 ±  212.247 ) / 44

Two real solutions:

 x =(163+√45049)/44= 8.528

or:

 x =(163-√45049)/44=-1.119

Two solutions were found :

1.  x =(163-√45049)/44=-1.119
2.  x =(163+√45049)/44= 8.528