Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((23•33x2) -  793x) +  216  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  216x²-793x+216 

The first term is,  216x²  its coefficient is  216 .
The middle term is,  -793x  its coefficient is  -793 .
The last term, "the constant", is  +216 

Step-1 : Multiply the coefficient of the first term by the constant   216 • 216 = 46656 

Step-2 : Find two factors of  46656  whose sum equals the coefficient of the middle term, which is   -793 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -729  and  -64 
                     216x² - 729x - 64x - 216

Step-4 : Add up the first 2 terms, pulling out like factors :
                    27x • (8x-27)
              Add up the last 2 terms, pulling out common factors :
                    8 • (8x-27)
Step-5 : Add up the four terms of step 4 :
                    (27x-8)  •  (8x-27)
             Which is the desired factorization

Equation at the end of step  2  :

  (8x - 27) • (27x - 8)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 3.2      Solve  :    8x-27 = 0 

 Add  27  to both sides of the equation : 
                      8x = 27
Divide both sides of the equation by 8:
                     x = 27/8 = 3.375

Solving a Single Variable Equation :

 3.3      Solve  :    27x-8 = 0 

 Add  8  to both sides of the equation : 
                      27x = 8
Divide both sides of the equation by 27:
                     x = 8/27 = 0.296

Supplement : Solving Quadratic Equation Directly

Solving    216x2-793x+216  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = 216x²-793x+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, 216 , 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   1.8356  

 Plugging into the parabola formula   1.8356  for  x  we can calculate the  y -coordinate : 
  y = 216.0 * 1.84 * 1.84 - 793.0 * 1.84 + 216.0
or   y = -511.834

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 216x²-793x+216
Axis of Symmetry (dashed)  {x}={ 1.84} 
Vertex at  {x,y} = { 1.84,-511.83} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.30, 0.00} 
Root 2 at  {x,y} = { 3.38, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   216x²-793x+216 = 0 by Completing The Square .

 Divide both sides of the equation by  216  to have 1 as the coefficient of the first term :
   x²-(793/216)x+1 = 0

Subtract  1  from both side of the equation :
   x²-(793/216)x = -1

Now the clever bit: Take the coefficient of  x , which is  793/216 , divide by two, giving  793/432 , and finally square it giving  793/432 

Add  793/432  to both sides of the equation :
  On the right hand side we have :
   -1  +  793/432    or,  (-1/1)+(793/432) 
  The common denominator of the two fractions is  432   Adding  (-432/432)+(793/432)  gives  361/432 
  So adding to both sides we finally get :
   x²-(793/216)x+(793/432) = 361/432

Adding  793/432  has completed the left hand side into a perfect square :
   x²-(793/216)x+(793/432)  =
   (x-(793/432)) • (x-(793/432))  =
  (x-(793/432))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(793/216)x+(793/432) = 361/432 and
   x²-(793/216)x+(793/432) = (x-(793/432))²
then, according to the law of transitivity,
   (x-(793/432))² = 361/432

We'll refer to this Equation as  Eq. #4.2.1  

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

Note that the square root of
   (x-(793/432))²   is
   (x-(793/432))^2/2 =
  (x-(793/432))¹ =
   x-(793/432)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x-(793/432) = √ 361/432

Add  793/432  to both sides to obtain:
   x = 793/432 + √ 361/432

Since a square root has two values, one positive and the other negative
   x² - (793/216)x + 1 = 0
   has two solutions:
  x = 793/432 + √ 361/432
   or
  x = 793/432 - √ 361/432

Note that  √ 361/432 can be written as
  √ 361  / √ 432   which is 19 / √ 432 

It is customary to further simplify until the denominator is radical free.

This can be achieved here by multiplying both the nominator and the denominator by  √ 432 

Following this multiplication, the numeric value of   19 /√ 432 remains unchanged, as it is multiplyed by  √ 432  / √ 432  which equals   1 

   OK, let's do it:

  19 •  √ 432       19 •  √ 432 
————————————————— = ————————————————
√ 432  • √ 432            432

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    216x²-793x+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   =    216
                      B   =   -793
                      C   =  216

Accordingly,  B²  -  4AC   =
                     628849 - 186624 =
                     442225

Applying the quadratic formula :

               793 ± √ 442225
   x  =    ————————
                        432

Can  √ 442225 be simplified ?

Yes!   The prime factorization of  442225   is
   5•5•7•7•19•19 
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).

√ 442225   =  √ 5•5•7•7•19•19   =5•7•19•√ 1   =
                ±  665 • √ 1   =
                ±  665

So now we are looking at:
           x  =  ( 793 ± 665) / 432

Two real solutions:

x =(793+√442225)/432=(793+665)/432= 3.375

or:

x =(793-√442225)/432=(793-665)/432= 0.296

Two solutions were found :

1.  x = 8/27 = 0.296
   
2.  x = 27/8 = 3.375