Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((0 -  24x2) +  197x) +  84  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  -16x²+197x+84 

The first term is,  -16x²  its coefficient is  -16 .
The middle term is,  +197x  its coefficient is  197 .
The last term, "the constant", is  +84 

Step-1 : Multiply the coefficient of the first term by the constant   -16 • 84 = -1344 

Step-2 : Find two factors of  -1344  whose sum equals the coefficient of the middle term, which is   197 .

For tidiness, printing of 22 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  :

  -16x2 + 197x + 84  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = -16x²+197x+84

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -16 , is negative (smaller 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   6.1562  

 Plugging into the parabola formula   6.1562  for  x  we can calculate the  y -coordinate : 
  y = -16.0 * 6.16 * 6.16 + 197.0 * 6.16 + 84.0
or   y = 690.391

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -16x²+197x+84
Axis of Symmetry (dashed)  {x}={ 6.16} 
Vertex at  {x,y} = { 6.16,690.39} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {12.73, 0.00} 
Root 2 at  {x,y} = {-0.41, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   -16x²+197x+84 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 16x²-197x-84 = 0  Divide both sides of the equation by  16  to have 1 as the coefficient of the first term :
   x²-(197/16)x-(21/4) = 0

Add  21/4  to both side of the equation :
   x²-(197/16)x = 21/4

Now the clever bit: Take the coefficient of  x , which is  197/16 , divide by two, giving  197/32 , and finally square it giving  38809/1024 

Add  38809/1024  to both sides of the equation :
  On the right hand side we have :
   21/4  +  38809/1024   The common denominator of the two fractions is  1024   Adding  (5376/1024)+(38809/1024)  gives  44185/1024 
  So adding to both sides we finally get :
   x²-(197/16)x+(38809/1024) = 44185/1024

Adding  38809/1024  has completed the left hand side into a perfect square :
   x²-(197/16)x+(38809/1024)  =
   (x-(197/32)) • (x-(197/32))  =
  (x-(197/32))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(197/16)x+(38809/1024) = 44185/1024 and
   x²-(197/16)x+(38809/1024) = (x-(197/32))²
then, according to the law of transitivity,
   (x-(197/32))² = 44185/1024

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-(197/32))²   is
   (x-(197/32))^2/2 =
  (x-(197/32))¹ =
   x-(197/32)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(197/32) = √ 44185/1024

Add  197/32  to both sides to obtain:
   x = 197/32 + √ 44185/1024

Since a square root has two values, one positive and the other negative
   x² - (197/16)x - (21/4) = 0
   has two solutions:
  x = 197/32 + √ 44185/1024
   or
  x = 197/32 - √ 44185/1024

Note that  √ 44185/1024 can be written as
  √ 44185  / √ 1024   which is √ 44185  / 32

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    -16x²+197x+84 = 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   =    -16
                      B   =   197
                      C   =   84

Accordingly,  B²  -  4AC   =
                     38809 - (-5376) =
                     44185

Applying the quadratic formula :

               -197 ± √ 44185
   x  =    ————————
                        -32

  √ 44185   , rounded to 4 decimal digits, is  210.2023
 So now we are looking at:
           x  =  ( -197 ±  210.202 ) / -32

Two real solutions:

 x =(-197+√44185)/-32=-0.413

or:

 x =(-197-√44185)/-32=12.725

Two solutions were found :

1.  x =(-197-√44185)/-32=12.725
2.  x =(-197+√44185)/-32=-0.413