Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (23d2 -  9d) +  9  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  8d²-9d+9 

The first term is,  8d²  its coefficient is  8 .
The middle term is,  -9d  its coefficient is  -9 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   8 • 9 = 72 

Step-2 : Find two factors of  72  whose sum equals the coefficient of the middle term, which is   -9 .

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  :

  8d2 - 9d + 9  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 8d²-9d+9

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, 8 , 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,Ad²+Bd+C,the  d -coordinate of the vertex is given by  -B/(2A) . In our case the  d  coordinate is   0.5625  

 Plugging into the parabola formula   0.5625  for  d  we can calculate the  y -coordinate : 
  y = 8.0 * 0.56 * 0.56 - 9.0 * 0.56 + 9.0
or   y = 6.469

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 8d²-9d+9
Axis of Symmetry (dashed)  {d}={ 0.56} 
Vertex at  {d,y} = { 0.56, 6.47} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 3.2     Solving   8d²-9d+9 = 0 by Completing The Square .

 Divide both sides of the equation by  8  to have 1 as the coefficient of the first term :
   d²-(9/8)d+(9/8) = 0

Subtract  9/8  from both side of the equation :
   d²-(9/8)d = -9/8

Now the clever bit: Take the coefficient of  d , which is  9/8 , divide by two, giving  9/16 , and finally square it giving  81/256 

Add  81/256  to both sides of the equation :
  On the right hand side we have :
   -9/8  +  81/256   The common denominator of the two fractions is  256   Adding  (-288/256)+(81/256)  gives  -207/256 
  So adding to both sides we finally get :
   d²-(9/8)d+(81/256) = -207/256

Adding  81/256  has completed the left hand side into a perfect square :
   d²-(9/8)d+(81/256)  =
   (d-(9/16)) • (d-(9/16))  =
  (d-(9/16))²
Things which are equal to the same thing are also equal to one another. Since
   d²-(9/8)d+(81/256) = -207/256 and
   d²-(9/8)d+(81/256) = (d-(9/16))²
then, according to the law of transitivity,
   (d-(9/16))² = -207/256

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
   (d-(9/16))²   is
   (d-(9/16))^2/2 =
  (d-(9/16))¹ =
   d-(9/16)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   d-(9/16) = √ -207/256

Add  9/16  to both sides to obtain:
   d = 9/16 + √ -207/256
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
   d² - (9/8)d + (9/8) = 0
   has two solutions:
  d = 9/16 + √ 207/256 •  i 
   or
  d = 9/16 - √ 207/256 •  i 

Note that  √ 207/256 can be written as
  √ 207  / √ 256   which is √ 207  / 16

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    8d²-9d+9 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  d  , the solution for   Ad²+Bd+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  d =   ————————
                      2A

  In our case,  A   =     8
                      B   =    -9
                      C   =   9

Accordingly,  B²  -  4AC   =
                     81 - 288 =
                     -207

Applying the quadratic formula :

               9 ± √ -207
   d  =    ——————
                      16

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,√ -207  = 
                    √ 207 • (-1)  =
                    √ 207  • √ -1   =
                    ±  √ 207  • i

Can  √ 207 be simplified ?

Yes!   The prime factorization of  207   is
   3•3•23 
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).

√ 207   =  √ 3•3•23   =
                ±  3 • √ 23

  √ 23   , rounded to 4 decimal digits, is   4.7958
 So now we are looking at:
           d  =  ( 9 ± 3 •  4.796 i ) / 16

Two imaginary solutions :

 d =(9+√-207)/16=(9+3i√ 23 )/16= 0.5625+0.8992i
  or: 
 d =(9-√-207)/16=(9-3i√ 23 )/16= 0.5625-0.8992i

Two solutions were found :

1.  d =(9-√-207)/16=(9-3i√ 23 )/16= 0.5625-0.8992i
2.  d =(9+√-207)/16=(9+3i√ 23 )/16= 0.5625+0.8992i