Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²+210x-96 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +210x  its coefficient is  210 .
The last term, "the constant", is  -96 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -96 = -96 

Step-2 : Find two factors of  -96  whose sum equals the coefficient of the middle term, which is   210 .

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

Equation at the end of step  1  :

  x2 + 210x - 96  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x²+210x-96

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, 1 , 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  -105.0000  

 Plugging into the parabola formula  -105.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * -105.00 * -105.00 + 210.0 * -105.00 - 96.0
or   y = -11121.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x²+210x-96
Axis of Symmetry (dashed)  {x}={-105.00} 
Vertex at  {x,y} = {-105.00,-11121.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-210.46, 0.00} 
Root 2 at  {x,y} = { 0.46, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x²+210x-96 = 0 by Completing The Square .

 Add  96  to both side of the equation :
   x²+210x = 96

Now the clever bit: Take the coefficient of  x , which is  210 , divide by two, giving  105 , and finally square it giving  11025 

Add  11025  to both sides of the equation :
  On the right hand side we have :
   96  +  11025    or,  (96/1)+(11025/1) 
  The common denominator of the two fractions is  1   Adding  (96/1)+(11025/1)  gives  11121/1 
  So adding to both sides we finally get :
   x²+210x+11025 = 11121

Adding  11025  has completed the left hand side into a perfect square :
   x²+210x+11025  =
   (x+105) • (x+105)  =
  (x+105)²
Things which are equal to the same thing are also equal to one another. Since
   x²+210x+11025 = 11121 and
   x²+210x+11025 = (x+105)²
then, according to the law of transitivity,
   (x+105)² = 11121

We'll refer to this Equation as  Eq. #2.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x+105)²   is
   (x+105)^2/2 =
  (x+105)¹ =
   x+105

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x+105 = √ 11121

Subtract  105  from both sides to obtain:
   x = -105 + √ 11121

Since a square root has two values, one positive and the other negative
   x² + 210x - 96 = 0
   has two solutions:
  x = -105 + √ 11121
   or
  x = -105 - √ 11121

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x²+210x-96 = 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   =     1
                      B   =   210
                      C   =  -96

Accordingly,  B²  -  4AC   =
                     44100 - (-384) =
                     44484

Applying the quadratic formula :

               -210 ± √ 44484
   x  =    ————————
                        2

Can  √ 44484 be simplified ?

Yes!   The prime factorization of  44484   is
   2•2•3•11•337 
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).

√ 44484   =  √ 2•2•3•11•337   =
                ±  2 • √ 11121

  √ 11121   , rounded to 4 decimal digits, is  105.4562
 So now we are looking at:
           x  =  ( -210 ± 2 •  105.456 ) / 2

Two real solutions:

 x =(-210+√44484)/2=-105+√ 11121 = 0.456

or:

 x =(-210-√44484)/2=-105-√ 11121 = -210.456

Two solutions were found :

1.  x =(-210-√44484)/2=-105-√ 11121 = -210.456
2.  x =(-210+√44484)/2=-105+√ 11121 = 0.456