Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3n2 -  53n) -  224  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  3n²-53n-224 

The first term is,  3n²  its coefficient is  3 .
The middle term is,  -53n  its coefficient is  -53 .
The last term, "the constant", is  -224 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -224 = -672 

Step-2 : Find two factors of  -672  whose sum equals the coefficient of the middle term, which is   -53 .

For tidiness, printing of 18 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  :

  3n2 - 53n - 224  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 3n²-53n-224

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, 3 , 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,An²+Bn+C,the  n -coordinate of the vertex is given by  -B/(2A) . In our case the  n  coordinate is   8.8333  

 Plugging into the parabola formula   8.8333  for  n  we can calculate the  y -coordinate : 
  y = 3.0 * 8.83 * 8.83 - 53.0 * 8.83 - 224.0
or   y = -458.083

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 3n²-53n-224
Axis of Symmetry (dashed)  {n}={ 8.83} 
Vertex at  {n,y} = { 8.83,-458.08} 
 n -Intercepts (Roots) :
Root 1 at  {n,y} = {-3.52, 0.00} 
Root 2 at  {n,y} = {21.19, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   3n²-53n-224 = 0 by Completing The Square .

 Divide both sides of the equation by  3  to have 1 as the coefficient of the first term :
   n²-(53/3)n-(224/3) = 0

Add  224/3  to both side of the equation :
   n²-(53/3)n = 224/3

Now the clever bit: Take the coefficient of  n , which is  53/3 , divide by two, giving  53/6 , and finally square it giving  2809/36 

Add  2809/36  to both sides of the equation :
  On the right hand side we have :
   224/3  +  2809/36   The common denominator of the two fractions is  36   Adding  (2688/36)+(2809/36)  gives  5497/36 
  So adding to both sides we finally get :
   n²-(53/3)n+(2809/36) = 5497/36

Adding  2809/36  has completed the left hand side into a perfect square :
   n²-(53/3)n+(2809/36)  =
   (n-(53/6)) • (n-(53/6))  =
  (n-(53/6))²
Things which are equal to the same thing are also equal to one another. Since
   n²-(53/3)n+(2809/36) = 5497/36 and
   n²-(53/3)n+(2809/36) = (n-(53/6))²
then, according to the law of transitivity,
   (n-(53/6))² = 5497/36

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
   (n-(53/6))²   is
   (n-(53/6))^2/2 =
  (n-(53/6))¹ =
   n-(53/6)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   n-(53/6) = √ 5497/36

Add  53/6  to both sides to obtain:
   n = 53/6 + √ 5497/36

Since a square root has two values, one positive and the other negative
   n² - (53/3)n - (224/3) = 0
   has two solutions:
  n = 53/6 + √ 5497/36
   or
  n = 53/6 - √ 5497/36

Note that  √ 5497/36 can be written as
  √ 5497  / √ 36   which is √ 5497  / 6

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    3n²-53n-224 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  n  , the solution for   An²+Bn+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  n =   ————————
                      2A

  In our case,  A   =     3
                      B   =   -53
                      C   =  -224

Accordingly,  B²  -  4AC   =
                     2809 - (-2688) =
                     5497

Applying the quadratic formula :

               53 ± √ 5497
   n  =    ——————
                      6

  √ 5497   , rounded to 4 decimal digits, is  74.1418
 So now we are looking at:
           n  =  ( 53 ±  74.142 ) / 6

Two real solutions:

 n =(53+√5497)/6=21.190

or:

 n =(53-√5497)/6=-3.524

Two solutions were found :

1.  n =(53-√5497)/6=-3.524
2.  n =(53+√5497)/6=21.190