Solution - Quadratic equations

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   -x² + 350x - 6250  =   -1 • (x² - 350x + 6250) 

Trying to factor by splitting the middle term

 2.2     Factoring  x² - 350x + 6250 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -350x  its coefficient is  -350 .
The last term, "the constant", is  +6250 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 6250 = 6250 

Step-2 : Find two factors of  6250  whose sum equals the coefficient of the middle term, which is   -350 .

For tidiness, printing of 18 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  :

  -x2 + 350x - 6250  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = -x²+350x-6250

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  175.0000  

 Plugging into the parabola formula  175.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 175.00 * 175.00 + 350.0 * 175.00 - 6250.0
or   y = 24375.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+350x-6250
Axis of Symmetry (dashed)  {x}={175.00} 
Vertex at  {x,y} = {175.00,24375.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {331.12, 0.00} 
Root 2 at  {x,y} = {18.88, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   -x²+350x-6250 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-350x+6250 = 0  Subtract  6250  from both side of the equation :
   x²-350x = -6250

Now the clever bit: Take the coefficient of  x , which is  350 , divide by two, giving  175 , and finally square it giving  30625 

Add  30625  to both sides of the equation :
  On the right hand side we have :
   -6250  +  30625    or,  (-6250/1)+(30625/1) 
  The common denominator of the two fractions is  1   Adding  (-6250/1)+(30625/1)  gives  24375/1 
  So adding to both sides we finally get :
   x²-350x+30625 = 24375

Adding  30625  has completed the left hand side into a perfect square :
   x²-350x+30625  =
   (x-175) • (x-175)  =
  (x-175)²
Things which are equal to the same thing are also equal to one another. Since
   x²-350x+30625 = 24375 and
   x²-350x+30625 = (x-175)²
then, according to the law of transitivity,
   (x-175)² = 24375

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-175)²   is
   (x-175)^2/2 =
  (x-175)¹ =
   x-175

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-175 = √ 24375

Add  175  to both sides to obtain:
   x = 175 + √ 24375

Since a square root has two values, one positive and the other negative
   x² - 350x + 6250 = 0
   has two solutions:
  x = 175 + √ 24375
   or
  x = 175 - √ 24375

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    -x²+350x-6250 = 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   =   350
                      C   =  -6250

Accordingly,  B²  -  4AC   =
                     122500 - 25000 =
                     97500

Applying the quadratic formula :

               -350 ± √ 97500
   x  =    ————————
                        -2

Can  √ 97500 be simplified ?

Yes!   The prime factorization of  97500   is
   2•2•3•5•5•5•5•13 
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).

√ 97500   =  √ 2•2•3•5•5•5•5•13   =2•5•5•√ 39   =
                ±  50 • √ 39

  √ 39   , rounded to 4 decimal digits, is   6.2450
 So now we are looking at:
           x  =  ( -350 ± 50 •  6.245 ) / -2

Two real solutions:

 x =(-350+√97500)/-2=175-25√ 39 = 18.875

or:

 x =(-350-√97500)/-2=175+25√ 39 = 331.125

Two solutions were found :

1.  x =(-350-√97500)/-2=175+25√ 39 = 331.125
2.  x =(-350+√97500)/-2=175-25√ 39 = 18.875