Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  p²+210p-3000 

The first term is,  p²  its coefficient is  1 .
The middle term is,  +210p  its coefficient is  210 .
The last term, "the constant", is  -3000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -3000 = -3000 

Step-2 : Find two factors of  -3000  whose sum equals the coefficient of the middle term, which is   210 .

For tidiness, printing of 26 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  1  :

  p2 + 210p - 3000  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = p²+210p-3000

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,Ap²+Bp+C,the  p -coordinate of the vertex is given by  -B/(2A) . In our case the  p  coordinate is  -105.0000  

 Plugging into the parabola formula  -105.0000  for  p  we can calculate the  y -coordinate : 
  y = 1.0 * -105.00 * -105.00 + 210.0 * -105.00 - 3000.0
or   y = -14025.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = p²+210p-3000
Axis of Symmetry (dashed)  {p}={-105.00} 
Vertex at  {p,y} = {-105.00,-14025.00} 
 p -Intercepts (Roots) :
Root 1 at  {p,y} = {-223.43, 0.00} 
Root 2 at  {p,y} = {13.43, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   p²+210p-3000 = 0 by Completing The Square .

 Add  3000  to both side of the equation :
   p²+210p = 3000

Now the clever bit: Take the coefficient of  p , 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 :
   3000  +  11025    or,  (3000/1)+(11025/1) 
  The common denominator of the two fractions is  1   Adding  (3000/1)+(11025/1)  gives  14025/1 
  So adding to both sides we finally get :
   p²+210p+11025 = 14025

Adding  11025  has completed the left hand side into a perfect square :
   p²+210p+11025  =
   (p+105) • (p+105)  =
  (p+105)²
Things which are equal to the same thing are also equal to one another. Since
   p²+210p+11025 = 14025 and
   p²+210p+11025 = (p+105)²
then, according to the law of transitivity,
   (p+105)² = 14025

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
   (p+105)²   is
   (p+105)^2/2 =
  (p+105)¹ =
   p+105

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   p+105 = √ 14025

Subtract  105  from both sides to obtain:
   p = -105 + √ 14025

Since a square root has two values, one positive and the other negative
   p² + 210p - 3000 = 0
   has two solutions:
  p = -105 + √ 14025
   or
  p = -105 - √ 14025

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    p²+210p-3000 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  p  , the solution for   Ap²+Bp+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  p =   ————————
                      2A

  In our case,  A   =     1
                      B   =   210
                      C   =  -3000

Accordingly,  B²  -  4AC   =
                     44100 - (-12000) =
                     56100

Applying the quadratic formula :

               -210 ± √ 56100
   p  =    ————————
                        2

Can  √ 56100 be simplified ?

Yes!   The prime factorization of  56100   is
   2•2•3•5•5•11•17 
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).

√ 56100   =  √ 2•2•3•5•5•11•17   =2•5•√ 561   =
                ±  10 • √ 561

  √ 561   , rounded to 4 decimal digits, is  23.6854
 So now we are looking at:
           p  =  ( -210 ± 10 •  23.685 ) / 2

Two real solutions:

 p =(-210+√56100)/2=-105+5√ 561 = 13.427

or:

 p =(-210-√56100)/2=-105-5√ 561 = -223.427

Two solutions were found :

1.  p =(-210-√56100)/2=-105-5√ 561 = -223.427
2.  p =(-210+√56100)/2=-105+5√ 561 = 13.427