Solution - Quadratic equations

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   -y² + 56y - 192  =   -1 • (y² - 56y + 192) 

Trying to factor by splitting the middle term

 2.2     Factoring  y² - 56y + 192 

The first term is,  y²  its coefficient is  1 .
The middle term is,  -56y  its coefficient is  -56 .
The last term, "the constant", is  +192 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 192 = 192 

Step-2 : Find two factors of  192  whose sum equals the coefficient of the middle term, which is   -56 .

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  :

  -y2 + 56y - 192  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   t = -y²+56y-192

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  "t"  because the coefficient of the first term, -1 , 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,Ay²+By+C,the  y -coordinate of the vertex is given by  -B/(2A) . In our case the  y  coordinate is  28.0000  

 Plugging into the parabola formula  28.0000  for  y  we can calculate the  t -coordinate : 
  t = -1.0 * 28.00 * 28.00 + 56.0 * 28.00 - 192.0
or   t = 592.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  t = -y²+56y-192
Axis of Symmetry (dashed)  {y}={28.00} 
Vertex at  {y,t} = {28.00,592.00} 
 y -Intercepts (Roots) :
Root 1 at  {y,t} = {52.33, 0.00} 
Root 2 at  {y,t} = { 3.67, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   -y²+56y-192 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 y²-56y+192 = 0  Subtract  192  from both side of the equation :
   y²-56y = -192

Now the clever bit: Take the coefficient of  y , which is  56 , divide by two, giving  28 , and finally square it giving  784 

Add  784  to both sides of the equation :
  On the right hand side we have :
   -192  +  784    or,  (-192/1)+(784/1) 
  The common denominator of the two fractions is  1   Adding  (-192/1)+(784/1)  gives  592/1 
  So adding to both sides we finally get :
   y²-56y+784 = 592

Adding  784  has completed the left hand side into a perfect square :
   y²-56y+784  =
   (y-28) • (y-28)  =
  (y-28)²
Things which are equal to the same thing are also equal to one another. Since
   y²-56y+784 = 592 and
   y²-56y+784 = (y-28)²
then, according to the law of transitivity,
   (y-28)² = 592

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
   (y-28)²   is
   (y-28)^2/2 =
  (y-28)¹ =
   y-28

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   y-28 = √ 592

Add  28  to both sides to obtain:
   y = 28 + √ 592

Since a square root has two values, one positive and the other negative
   y² - 56y + 192 = 0
   has two solutions:
  y = 28 + √ 592
   or
  y = 28 - √ 592

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    -y²+56y-192 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  y  , the solution for   Ay²+By+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  y =   ————————
                      2A

  In our case,  A   =     -1
                      B   =    56
                      C   =  -192

Accordingly,  B²  -  4AC   =
                     3136 - 768 =
                     2368

Applying the quadratic formula :

               -56 ± √ 2368
   y  =    ———————
                        -2

Can  √ 2368 be simplified ?

Yes!   The prime factorization of  2368   is
   2•2•2•2•2•2•37 
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).

√ 2368   =  √ 2•2•2•2•2•2•37   =2•2•2•√ 37   =
                ±  8 • √ 37

  √ 37   , rounded to 4 decimal digits, is   6.0828
 So now we are looking at:
           y  =  ( -56 ± 8 •  6.083 ) / -2

Two real solutions:

 y =(-56+√2368)/-2=28-4√ 37 = 3.669

or:

 y =(-56-√2368)/-2=28+4√ 37 = 52.331

Two solutions were found :

1.  y =(-56-√2368)/-2=28+4√ 37 = 52.331
2.  y =(-56+√2368)/-2=28-4√ 37 = 3.669