Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (11n2 -  9n) -  1  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  11n²-9n-1 

The first term is,  11n²  its coefficient is  11 .
The middle term is,  -9n  its coefficient is  -9 .
The last term, "the constant", is  -1 

Step-1 : Multiply the coefficient of the first term by the constant   11 • -1 = -11 

Step-2 : Find two factors of  -11  whose sum equals the coefficient of the middle term, which is   -9 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  2  :

  11n2 - 9n - 1  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 11n²-9n-1

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, 11 , 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   0.4091  

 Plugging into the parabola formula   0.4091  for  n  we can calculate the  y -coordinate : 
  y = 11.0 * 0.41 * 0.41 - 9.0 * 0.41 - 1.0
or   y = -2.841

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 11n²-9n-1
Axis of Symmetry (dashed)  {n}={ 0.41} 
Vertex at  {n,y} = { 0.41,-2.84} 
 n -Intercepts (Roots) :
Root 1 at  {n,y} = {-0.10, 0.00} 
Root 2 at  {n,y} = { 0.92, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   11n²-9n-1 = 0 by Completing The Square .

 Divide both sides of the equation by  11  to have 1 as the coefficient of the first term :
   n²-(9/11)n-(1/11) = 0

Add  1/11  to both side of the equation :
   n²-(9/11)n = 1/11

Now the clever bit: Take the coefficient of  n , which is  9/11 , divide by two, giving  9/22 , and finally square it giving  81/484 

Add  81/484  to both sides of the equation :
  On the right hand side we have :
   1/11  +  81/484   The common denominator of the two fractions is  484   Adding  (44/484)+(81/484)  gives  125/484 
  So adding to both sides we finally get :
   n²-(9/11)n+(81/484) = 125/484

Adding  81/484  has completed the left hand side into a perfect square :
   n²-(9/11)n+(81/484)  =
   (n-(9/22)) • (n-(9/22))  =
  (n-(9/22))²
Things which are equal to the same thing are also equal to one another. Since
   n²-(9/11)n+(81/484) = 125/484 and
   n²-(9/11)n+(81/484) = (n-(9/22))²
then, according to the law of transitivity,
   (n-(9/22))² = 125/484

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-(9/22))²   is
   (n-(9/22))^2/2 =
  (n-(9/22))¹ =
   n-(9/22)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   n-(9/22) = √ 125/484

Add  9/22  to both sides to obtain:
   n = 9/22 + √ 125/484

Since a square root has two values, one positive and the other negative
   n² - (9/11)n - (1/11) = 0
   has two solutions:
  n = 9/22 + √ 125/484
   or
  n = 9/22 - √ 125/484

Note that  √ 125/484 can be written as
  √ 125  / √ 484   which is √ 125  / 22

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    11n²-9n-1 = 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   =     11
                      B   =    -9
                      C   =   -1

Accordingly,  B²  -  4AC   =
                     81 - (-44) =
                     125

Applying the quadratic formula :

               9 ± √ 125
   n  =    —————
                    22

Can  √ 125 be simplified ?

Yes!   The prime factorization of  125   is
   5•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).

√ 125   =  √ 5•5•5   =
                ±  5 • √ 5

  √ 5   , rounded to 4 decimal digits, is   2.2361
 So now we are looking at:
           n  =  ( 9 ± 5 •  2.236 ) / 22

Two real solutions:

 n =(9+√125)/22=(9+5√ 5 )/22= 0.917

or:

 n =(9-√125)/22=(9-5√ 5 )/22= -0.099

Two solutions were found :

1.  n =(9-√125)/22=(9-5√ 5 )/22= -0.099
2.  n =(9+√125)/22=(9+5√ 5 )/22= 0.917