Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  z²-2z+2 

The first term is,  z²  its coefficient is  1 .
The middle term is,  -2z  its coefficient is  -2 .
The last term, "the constant", is  +2 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 2 = 2 

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

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

Equation at the end of step  1  :

  z2 - 2z + 2  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = z²-2z+2

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,Az²+Bz+C,the  z -coordinate of the vertex is given by  -B/(2A) . In our case the  z  coordinate is   1.0000  

 Plugging into the parabola formula   1.0000  for  z  we can calculate the  y -coordinate : 
  y = 1.0 * 1.00 * 1.00 - 2.0 * 1.00 + 2.0
or   y = 1.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = z²-2z+2
Axis of Symmetry (dashed)  {z}={ 1.00} 
Vertex at  {z,y} = { 1.00, 1.00} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 2.2     Solving   z²-2z+2 = 0 by Completing The Square .

 Subtract  2  from both side of the equation :
   z²-2z = -2

Now the clever bit: Take the coefficient of  z , which is  2 , divide by two, giving  1 , and finally square it giving  1 

Add  1  to both sides of the equation :
  On the right hand side we have :
   -2  +  1    or,  (-2/1)+(1/1) 
  The common denominator of the two fractions is  1   Adding  (-2/1)+(1/1)  gives  -1/1 
  So adding to both sides we finally get :
   z²-2z+1 = -1

Adding  1  has completed the left hand side into a perfect square :
   z²-2z+1  =
   (z-1) • (z-1)  =
  (z-1)²
Things which are equal to the same thing are also equal to one another. Since
   z²-2z+1 = -1 and
   z²-2z+1 = (z-1)²
then, according to the law of transitivity,
   (z-1)² = -1

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
   (z-1)²   is
   (z-1)^2/2 =
  (z-1)¹ =
   z-1

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   z-1 = √ -1

Add  1  to both sides to obtain:
   z = 1 + √ -1
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
   z² - 2z + 2 = 0
   has two solutions:
  z = 1 + √ 1 •  i 
   or
  z = 1 - √ 1 •  i 

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    z²-2z+2 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  z  , the solution for   Az²+Bz+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  z =   ————————
                      2A

  In our case,  A   =     1
                      B   =    -2
                      C   =   2

Accordingly,  B²  -  4AC   =
                     4 - 8 =
                     -4

Applying the quadratic formula :

               2 ± √ -4
   z  =    —————
                    2

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,√ -4  = 
                    √ 4 • (-1)  =
                    √ 4  • √ -1   =
                    ±  √ 4  • i

Can  √ 4 be simplified ?

Yes!   The prime factorization of  4   is
   2•2 
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).

√ 4   =  √ 2•2   =
                ±  2 • √ 1   =
                ±  2

So now we are looking at:
           z  =  ( 2 ± 2i ) / 2

Two imaginary solutions :

 z =(2+√-4)/2=1+i= 1.0000+1.0000i
  or: 
 z =(2-√-4)/2=1-i= 1.0000-1.0000i

Two solutions were found :

1.  z =(2-√-4)/2=1-i= 1.0000-1.0000i
2.  z =(2+√-4)/2=1+i= 1.0000+1.0000i