Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  -x²+120x+5000 

The first term is,  -x²  its coefficient is  -1 .
The middle term is,  +120x  its coefficient is  120 .
The last term, "the constant", is  +5000 

Step-1 : Multiply the coefficient of the first term by the constant   -1 • 5000 = -5000 

Step-2 : Find two factors of  -5000  whose sum equals the coefficient of the middle term, which is   120 .

For tidiness, printing of 14 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  :

  -x2 + 120x + 5000  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = -x²+120x+5000

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, -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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is  60.0000  

 Plugging into the parabola formula  60.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 60.00 * 60.00 + 120.0 * 60.00 + 5000.0
or   y = 8600.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+120x+5000
Axis of Symmetry (dashed)  {x}={60.00} 
Vertex at  {x,y} = {60.00,8600.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {152.74, 0.00} 
Root 2 at  {x,y} = {-32.74, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   -x²+120x+5000 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-120x-5000 = 0  Add  5000  to both side of the equation :
   x²-120x = 5000

Now the clever bit: Take the coefficient of  x , which is  120 , divide by two, giving  60 , and finally square it giving  3600 

Add  3600  to both sides of the equation :
  On the right hand side we have :
   5000  +  3600    or,  (5000/1)+(3600/1) 
  The common denominator of the two fractions is  1   Adding  (5000/1)+(3600/1)  gives  8600/1 
  So adding to both sides we finally get :
   x²-120x+3600 = 8600

Adding  3600  has completed the left hand side into a perfect square :
   x²-120x+3600  =
   (x-60) • (x-60)  =
  (x-60)²
Things which are equal to the same thing are also equal to one another. Since
   x²-120x+3600 = 8600 and
   x²-120x+3600 = (x-60)²
then, according to the law of transitivity,
   (x-60)² = 8600

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-60)²   is
   (x-60)^2/2 =
  (x-60)¹ =
   x-60

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x-60 = √ 8600

Add  60  to both sides to obtain:
   x = 60 + √ 8600

Since a square root has two values, one positive and the other negative
   x² - 120x - 5000 = 0
   has two solutions:
  x = 60 + √ 8600
   or
  x = 60 - √ 8600

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    -x²+120x+5000 = 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   =   120
                      C   =  5000

Accordingly,  B²  -  4AC   =
                     14400 - (-20000) =
                     34400

Applying the quadratic formula :

               -120 ± √ 34400
   x  =    ————————
                        -2

Can  √ 34400 be simplified ?

Yes!   The prime factorization of  34400   is
   2•2•2•2•2•5•5•43 
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).

√ 34400   =  √ 2•2•2•2•2•5•5•43   =2•2•5•√ 86   =
                ±  20 • √ 86

  √ 86   , rounded to 4 decimal digits, is   9.2736
 So now we are looking at:
           x  =  ( -120 ± 20 •  9.274 ) / -2

Two real solutions:

 x =(-120+√34400)/-2=60-10√ 86 = -32.736

or:

 x =(-120-√34400)/-2=60+10√ 86 = 152.736

Two solutions were found :

1.  x =(-120-√34400)/-2=60+10√ 86 = 152.736
2.  x =(-120+√34400)/-2=60-10√ 86 = -32.736