Solution - Quadratic equations

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     -11*a^2-5*a-(6)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((0 -  11a2) -  5a) -  6  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -11a² - 5a - 6  =   -1 • (11a² + 5a + 6) 

Trying to factor by splitting the middle term

 3.2     Factoring  11a² + 5a + 6 

The first term is,  11a²  its coefficient is  11 .
The middle term is,  +5a  its coefficient is  5 .
The last term, "the constant", is  +6 

Step-1 : Multiply the coefficient of the first term by the constant   11 • 6 = 66 

Step-2 : Find two factors of  66  whose sum equals the coefficient of the middle term, which is   5 .

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

Equation at the end of step  3  :

  -11a2 - 5a - 6  = 0 

Step  4  :

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = -11a²-5a-6

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -11 , is negative (smaller 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,Aa²+Ba+C,the  a -coordinate of the vertex is given by  -B/(2A) . In our case the  a  coordinate is  -0.2273  

 Plugging into the parabola formula  -0.2273  for  a  we can calculate the  y -coordinate : 
  y = -11.0 * -0.23 * -0.23 - 5.0 * -0.23 - 6.0
or   y = -5.432

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -11a²-5a-6
Axis of Symmetry (dashed)  {a}={-0.23} 
Vertex at  {a,y} = {-0.23,-5.43} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 4.2     Solving   -11a²-5a-6 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 11a²+5a+6 = 0  Divide both sides of the equation by  11  to have 1 as the coefficient of the first term :
   a²+(5/11)a+(6/11) = 0

Subtract  6/11  from both side of the equation :
   a²+(5/11)a = -6/11

Now the clever bit: Take the coefficient of  a , which is  5/11 , divide by two, giving  5/22 , and finally square it giving  25/484 

Add  25/484  to both sides of the equation :
  On the right hand side we have :
   -6/11  +  25/484   The common denominator of the two fractions is  484   Adding  (-264/484)+(25/484)  gives  -239/484 
  So adding to both sides we finally get :
   a²+(5/11)a+(25/484) = -239/484

Adding  25/484  has completed the left hand side into a perfect square :
   a²+(5/11)a+(25/484)  =
   (a+(5/22)) • (a+(5/22))  =
  (a+(5/22))²
Things which are equal to the same thing are also equal to one another. Since
   a²+(5/11)a+(25/484) = -239/484 and
   a²+(5/11)a+(25/484) = (a+(5/22))²
then, according to the law of transitivity,
   (a+(5/22))² = -239/484

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
   (a+(5/22))²   is
   (a+(5/22))^2/2 =
  (a+(5/22))¹ =
   a+(5/22)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   a+(5/22) = √ -239/484

Subtract  5/22  from both sides to obtain:
   a = -5/22 + √ -239/484
In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 


Since a square root has two values, one positive and the other negative
   a² + (5/11)a + (6/11) = 0
   has two solutions:
  a = -5/22 + √ 239/484 •  i 
   or
  a = -5/22 - √ 239/484 •  i 

Note that  √ 239/484 can be written as
  √ 239  / √ 484   which is √ 239  / 22

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    -11a²-5a-6 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  a  , the solution for   Aa²+Ba+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  a =   ————————
                      2A

  In our case,  A   =    -11
                      B   =    -5
                      C   =   -6

Accordingly,  B²  -  4AC   =
                     25 - 264 =
                     -239

Applying the quadratic formula :

               5 ± √ -239
   a  =    ——————
                      -22

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -239  = 
                    √ 239 • (-1)  =
                    √ 239  • √ -1   =
                    ±  √ 239  • i

  √ 239   , rounded to 4 decimal digits, is  15.4596
 So now we are looking at:
           a  =  ( 5 ±  15.460 i ) / -22

Two imaginary solutions :

 a =(5+√-239)/-22=5/-22-i/22√ 239 = -0.2273+0.7027i
  or: 
 a =(5-√-239)/-22=5/-22+i/22√ 239 = -0.2273-0.7027i

Two solutions were found :

1.  a =(5-√-239)/-22=5/-22+i/22√ 239 = -0.2273-0.7027i
2.  a =(5+√-239)/-22=5/-22-i/22√ 239 = -0.2273+0.7027i