Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (26x2 -  72x) -  57  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  64x²-72x-57 

The first term is,  64x²  its coefficient is  64 .
The middle term is,  -72x  its coefficient is  -72 .
The last term, "the constant", is  -57 

Step-1 : Multiply the coefficient of the first term by the constant   64 • -57 = -3648 

Step-2 : Find two factors of  -3648  whose sum equals the coefficient of the middle term, which is   -72 .

For tidiness, printing of 22 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  :

  64x2 - 72x - 57  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 64x²-72x-57

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, 64 , 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   0.5625  

 Plugging into the parabola formula   0.5625  for  x  we can calculate the  y -coordinate : 
  y = 64.0 * 0.56 * 0.56 - 72.0 * 0.56 - 57.0
or   y = -77.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 64x²-72x-57
Axis of Symmetry (dashed)  {x}={ 0.56} 
Vertex at  {x,y} = { 0.56,-77.25} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.54, 0.00} 
Root 2 at  {x,y} = { 1.66, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   64x²-72x-57 = 0 by Completing The Square .

 Divide both sides of the equation by  64  to have 1 as the coefficient of the first term :
   x²-(9/8)x-(57/64) = 0

Add  57/64  to both side of the equation :
   x²-(9/8)x = 57/64

Now the clever bit: Take the coefficient of  x , 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 :
   57/64  +  81/256   The common denominator of the two fractions is  256   Adding  (228/256)+(81/256)  gives  309/256 
  So adding to both sides we finally get :
   x²-(9/8)x+(81/256) = 309/256

Adding  81/256  has completed the left hand side into a perfect square :
   x²-(9/8)x+(81/256)  =
   (x-(9/16)) • (x-(9/16))  =
  (x-(9/16))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(9/8)x+(81/256) = 309/256 and
   x²-(9/8)x+(81/256) = (x-(9/16))²
then, according to the law of transitivity,
   (x-(9/16))² = 309/256

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

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(9/16) = √ 309/256

Add  9/16  to both sides to obtain:
   x = 9/16 + √ 309/256

Since a square root has two values, one positive and the other negative
   x² - (9/8)x - (57/64) = 0
   has two solutions:
  x = 9/16 + √ 309/256
   or
  x = 9/16 - √ 309/256

Note that  √ 309/256 can be written as
  √ 309  / √ 256   which is √ 309  / 16

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    64x²-72x-57 = 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   =     64
                      B   =   -72
                      C   =  -57

Accordingly,  B²  -  4AC   =
                     5184 - (-14592) =
                     19776

Applying the quadratic formula :

               72 ± √ 19776
   x  =    ———————
                        128

Can  √ 19776 be simplified ?

Yes!   The prime factorization of  19776   is
   2•2•2•2•2•2•3•103 
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).

√ 19776   =  √ 2•2•2•2•2•2•3•103   =2•2•2•√ 309   =
                ±  8 • √ 309

  √ 309   , rounded to 4 decimal digits, is  17.5784
 So now we are looking at:
           x  =  ( 72 ± 8 •  17.578 ) / 128

Two real solutions:

 x =(72+√19776)/128=(9+√ 309 )/16= 1.661

or:

 x =(72-√19776)/128=(9-√ 309 )/16= -0.536

Two solutions were found :

1.  x =(72-√19776)/128=(9-√ 309 )/16= -0.536
2.  x =(72+√19776)/128=(9+√ 309 )/16= 1.661