Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((0 -  24x2) +  146x) +  137  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  -16x²+146x+137 

The first term is,  -16x²  its coefficient is  -16 .
The middle term is,  +146x  its coefficient is  146 .
The last term, "the constant", is  +137 

Step-1 : Multiply the coefficient of the first term by the constant   -16 • 137 = -2192 

Step-2 : Find two factors of  -2192  whose sum equals the coefficient of the middle term, which is   146 .

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

Equation at the end of step  2  :

  -16x2 + 146x + 137  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = -16x²+146x+137

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, -16 , 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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   4.5625  

 Plugging into the parabola formula   4.5625  for  x  we can calculate the  y -coordinate : 
  y = -16.0 * 4.56 * 4.56 + 146.0 * 4.56 + 137.0
or   y = 470.062

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -16x²+146x+137
Axis of Symmetry (dashed)  {x}={ 4.56} 
Vertex at  {x,y} = { 4.56,470.06} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 9.98, 0.00} 
Root 2 at  {x,y} = {-0.86, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   -16x²+146x+137 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 16x²-146x-137 = 0  Divide both sides of the equation by  16  to have 1 as the coefficient of the first term :
   x²-(73/8)x-(137/16) = 0

Add  137/16  to both side of the equation :
   x²-(73/8)x = 137/16

Now the clever bit: Take the coefficient of  x , which is  73/8 , divide by two, giving  73/16 , and finally square it giving  5329/256 

Add  5329/256  to both sides of the equation :
  On the right hand side we have :
   137/16  +  5329/256   The common denominator of the two fractions is  256   Adding  (2192/256)+(5329/256)  gives  7521/256 
  So adding to both sides we finally get :
   x²-(73/8)x+(5329/256) = 7521/256

Adding  5329/256  has completed the left hand side into a perfect square :
   x²-(73/8)x+(5329/256)  =
   (x-(73/16)) • (x-(73/16))  =
  (x-(73/16))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(73/8)x+(5329/256) = 7521/256 and
   x²-(73/8)x+(5329/256) = (x-(73/16))²
then, according to the law of transitivity,
   (x-(73/16))² = 7521/256

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
   (x-(73/16))²   is
   (x-(73/16))^2/2 =
  (x-(73/16))¹ =
   x-(73/16)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(73/16) = √ 7521/256

Add  73/16  to both sides to obtain:
   x = 73/16 + √ 7521/256

Since a square root has two values, one positive and the other negative
   x² - (73/8)x - (137/16) = 0
   has two solutions:
  x = 73/16 + √ 7521/256
   or
  x = 73/16 - √ 7521/256

Note that  √ 7521/256 can be written as
  √ 7521  / √ 256   which is √ 7521  / 16

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    -16x²+146x+137 = 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   =    -16
                      B   =   146
                      C   =  137

Accordingly,  B²  -  4AC   =
                     21316 - (-8768) =
                     30084

Applying the quadratic formula :

               -146 ± √ 30084
   x  =    ————————
                        -32

Can  √ 30084 be simplified ?

Yes!   The prime factorization of  30084   is
   2•2•3•23•109 
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).

√ 30084   =  √ 2•2•3•23•109   =
                ±  2 • √ 7521

  √ 7521   , rounded to 4 decimal digits, is  86.7237
 So now we are looking at:
           x  =  ( -146 ± 2 •  86.724 ) / -32

Two real solutions:

 x =(-146+√30084)/-32=(73-√ 7521 )/16= -0.858

or:

 x =(-146-√30084)/-32=(73+√ 7521 )/16= 9.983

Two solutions were found :

1.  x =(-146-√30084)/-32=(73+√ 7521 )/16= 9.983
2.  x =(-146+√30084)/-32=(73-√ 7521 )/16= -0.858