Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((25•3x2) -  7x) -  20  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  96x²-7x-20 

The first term is,  96x²  its coefficient is  96 .
The middle term is,  -7x  its coefficient is  -7 .
The last term, "the constant", is  -20 

Step-1 : Multiply the coefficient of the first term by the constant   96 • -20 = -1920 

Step-2 : Find two factors of  -1920  whose sum equals the coefficient of the middle term, which is   -7 .

For tidiness, printing of 26 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  :

  96x2 - 7x - 20  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 96x²-7x-20

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, 96 , 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   0.0365  

 Plugging into the parabola formula   0.0365  for  x  we can calculate the  y -coordinate : 
  y = 96.0 * 0.04 * 0.04 - 7.0 * 0.04 - 20.0
or   y = -20.128

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 96x²-7x-20
Axis of Symmetry (dashed)  {x}={ 0.04} 
Vertex at  {x,y} = { 0.04,-20.13} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.42, 0.00} 
Root 2 at  {x,y} = { 0.49, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   96x²-7x-20 = 0 by Completing The Square .

 Divide both sides of the equation by  96  to have 1 as the coefficient of the first term :
   x²-(7/96)x-(5/24) = 0

Add  5/24  to both side of the equation :
   x²-(7/96)x = 5/24

Now the clever bit: Take the coefficient of  x , which is  7/96 , divide by two, giving  7/192 , and finally square it giving  49/36864 

Add  49/36864  to both sides of the equation :
  On the right hand side we have :
   5/24  +  49/36864   The common denominator of the two fractions is  36864   Adding  (7680/36864)+(49/36864)  gives  7729/36864 
  So adding to both sides we finally get :
   x²-(7/96)x+(49/36864) = 7729/36864

Adding  49/36864  has completed the left hand side into a perfect square :
   x²-(7/96)x+(49/36864)  =
   (x-(7/192)) • (x-(7/192))  =
  (x-(7/192))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(7/96)x+(49/36864) = 7729/36864 and
   x²-(7/96)x+(49/36864) = (x-(7/192))²
then, according to the law of transitivity,
   (x-(7/192))² = 7729/36864

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-(7/192))²   is
   (x-(7/192))^2/2 =
  (x-(7/192))¹ =
   x-(7/192)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(7/192) = √ 7729/36864

Add  7/192  to both sides to obtain:
   x = 7/192 + √ 7729/36864

Since a square root has two values, one positive and the other negative
   x² - (7/96)x - (5/24) = 0
   has two solutions:
  x = 7/192 + √ 7729/36864
   or
  x = 7/192 - √ 7729/36864

Note that  √ 7729/36864 can be written as
  √ 7729  / √ 36864   which is √ 7729  / 192

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    96x²-7x-20 = 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   =     96
                      B   =    -7
                      C   =  -20

Accordingly,  B²  -  4AC   =
                     49 - (-7680) =
                     7729

Applying the quadratic formula :

               7 ± √ 7729
   x  =    ——————
                      192

  √ 7729   , rounded to 4 decimal digits, is  87.9147
 So now we are looking at:
           x  =  ( 7 ±  87.915 ) / 192

Two real solutions:

 x =(7+√7729)/192= 0.494

or:

 x =(7-√7729)/192=-0.421

Two solutions were found :

1.  x =(7-√7729)/192=-0.421
2.  x =(7+√7729)/192= 0.494