Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²-108x+216 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -108x  its coefficient is  -108 .
The last term, "the constant", is  +216 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 216 = 216 

Step-2 : Find two factors of  216  whose sum equals the coefficient of the middle term, which is   -108 .

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  1  :

  x2 - 108x + 216  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x²-108x+216

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, 1 , 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  54.0000  

 Plugging into the parabola formula  54.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 54.00 * 54.00 - 108.0 * 54.00 + 216.0
or   y = -2700.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x²-108x+216
Axis of Symmetry (dashed)  {x}={54.00} 
Vertex at  {x,y} = {54.00,-2700.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 2.04, 0.00} 
Root 2 at  {x,y} = {105.96, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x²-108x+216 = 0 by Completing The Square .

 Subtract  216  from both side of the equation :
   x²-108x = -216

Now the clever bit: Take the coefficient of  x , which is  108 , divide by two, giving  54 , and finally square it giving  2916 

Add  2916  to both sides of the equation :
  On the right hand side we have :
   -216  +  2916    or,  (-216/1)+(2916/1) 
  The common denominator of the two fractions is  1   Adding  (-216/1)+(2916/1)  gives  2700/1 
  So adding to both sides we finally get :
   x²-108x+2916 = 2700

Adding  2916  has completed the left hand side into a perfect square :
   x²-108x+2916  =
   (x-54) • (x-54)  =
  (x-54)²
Things which are equal to the same thing are also equal to one another. Since
   x²-108x+2916 = 2700 and
   x²-108x+2916 = (x-54)²
then, according to the law of transitivity,
   (x-54)² = 2700

We'll refer to this Equation as  Eq. #2.2.1  

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

Note that the square root of
   (x-54)²   is
   (x-54)^2/2 =
  (x-54)¹ =
   x-54

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x-54 = √ 2700

Add  54  to both sides to obtain:
   x = 54 + √ 2700

Since a square root has two values, one positive and the other negative
   x² - 108x + 216 = 0
   has two solutions:
  x = 54 + √ 2700
   or
  x = 54 - √ 2700

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x²-108x+216 = 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   =     1
                      B   =   -108
                      C   =  216

Accordingly,  B²  -  4AC   =
                     11664 - 864 =
                     10800

Applying the quadratic formula :

               108 ± √ 10800
   x  =    ————————
                        2

Can  √ 10800 be simplified ?

Yes!   The prime factorization of  10800   is
   2•2•2•2•3•3•3•5•5 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 10800   =  √ 2•2•2•2•3•3•3•5•5   =2•2•3•5•√ 3   =
                ±  60 • √ 3

  √ 3   , rounded to 4 decimal digits, is   1.7321
 So now we are looking at:
           x  =  ( 108 ± 60 •  1.732 ) / 2

Two real solutions:

 x =(108+√10800)/2=54+30√ 3 = 105.962

or:

 x =(108-√10800)/2=54-30√ 3 = 2.038

Two solutions were found :

1.  x =(108-√10800)/2=54-30√ 3 = 2.038
2.  x =(108+√10800)/2=54+30√ 3 = 105.962