Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•32x2) -  9x) -  38  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  18x²-9x-38 

The first term is,  18x²  its coefficient is  18 .
The middle term is,  -9x  its coefficient is  -9 .
The last term, "the constant", is  -38 

Step-1 : Multiply the coefficient of the first term by the constant   18 • -38 = -684 

Step-2 : Find two factors of  -684  whose sum equals the coefficient of the middle term, which is   -9 .

For tidiness, printing of 12 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  :

  18x2 - 9x - 38  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 18x²-9x-38

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, 18 , 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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   0.2500  

 Plugging into the parabola formula   0.2500  for  x  we can calculate the  y -coordinate : 
  y = 18.0 * 0.25 * 0.25 - 9.0 * 0.25 - 38.0
or   y = -39.125

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 18x²-9x-38
Axis of Symmetry (dashed)  {x}={ 0.25} 
Vertex at  {x,y} = { 0.25,-39.12} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.22, 0.00} 
Root 2 at  {x,y} = { 1.72, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   18x²-9x-38 = 0 by Completing The Square .

 Divide both sides of the equation by  18  to have 1 as the coefficient of the first term :
   x²-(1/2)x-(19/9) = 0

Add  19/9  to both side of the equation :
   x²-(1/2)x = 19/9

Now the clever bit: Take the coefficient of  x , which is  1/2 , divide by two, giving  1/4 , and finally square it giving  1/16 

Add  1/16  to both sides of the equation :
  On the right hand side we have :
   19/9  +  1/16   The common denominator of the two fractions is  144   Adding  (304/144)+(9/144)  gives  313/144 
  So adding to both sides we finally get :
   x²-(1/2)x+(1/16) = 313/144

Adding  1/16  has completed the left hand side into a perfect square :
   x²-(1/2)x+(1/16)  =
   (x-(1/4)) • (x-(1/4))  =
  (x-(1/4))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(1/2)x+(1/16) = 313/144 and
   x²-(1/2)x+(1/16) = (x-(1/4))²
then, according to the law of transitivity,
   (x-(1/4))² = 313/144

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
   (x-(1/4))²   is
   (x-(1/4))^2/2 =
  (x-(1/4))¹ =
   x-(1/4)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(1/4) = √ 313/144

Add  1/4  to both sides to obtain:
   x = 1/4 + √ 313/144

Since a square root has two values, one positive and the other negative
   x² - (1/2)x - (19/9) = 0
   has two solutions:
  x = 1/4 + √ 313/144
   or
  x = 1/4 - √ 313/144

Note that  √ 313/144 can be written as
  √ 313  / √ 144   which is √ 313  / 12

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    18x²-9x-38 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax²+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  x =   ————————
                      2A

  In our case,  A   =     18
                      B   =    -9
                      C   =  -38

Accordingly,  B²  -  4AC   =
                     81 - (-2736) =
                     2817

Applying the quadratic formula :

               9 ± √ 2817
   x  =    ——————
                      36

Can  √ 2817 be simplified ?

Yes!   The prime factorization of  2817   is
   3•3•313 
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).

√ 2817   =  √ 3•3•313   =
                ±  3 • √ 313

  √ 313   , rounded to 4 decimal digits, is  17.6918
 So now we are looking at:
           x  =  ( 9 ± 3 •  17.692 ) / 36

Two real solutions:

 x =(9+√2817)/36=1/4+1/12√ 313 = 1.724

or:

 x =(9-√2817)/36=1/4-1/12√ 313 = -1.224

Two solutions were found :

1.  x =(9-√2817)/36=1/4-1/12√ 313 = -1.224
2.  x =(9+√2817)/36=1/4+1/12√ 313 = 1.724