Solution - Quadratic equations

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   -x² + 270x - 1225  =   -1 • (x² - 270x + 1225) 

Trying to factor by splitting the middle term

 2.2     Factoring  x² - 270x + 1225 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -270x  its coefficient is  -270 .
The last term, "the constant", is  +1225 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1225 = 1225 

Step-2 : Find two factors of  1225  whose sum equals the coefficient of the middle term, which is   -270 .

For tidiness, printing of 12 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 + 270x - 1225  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = -x²+270x-1225

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  135.0000  

 Plugging into the parabola formula  135.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 135.00 * 135.00 + 270.0 * 135.00 - 1225.0
or   y = 17000.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+270x-1225
Axis of Symmetry (dashed)  {x}={135.00} 
Vertex at  {x,y} = {135.00,17000.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {265.38, 0.00} 
Root 2 at  {x,y} = { 4.62, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   -x²+270x-1225 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-270x+1225 = 0  Subtract  1225  from both side of the equation :
   x²-270x = -1225

Now the clever bit: Take the coefficient of  x , which is  270 , divide by two, giving  135 , and finally square it giving  18225 

Add  18225  to both sides of the equation :
  On the right hand side we have :
   -1225  +  18225    or,  (-1225/1)+(18225/1) 
  The common denominator of the two fractions is  1   Adding  (-1225/1)+(18225/1)  gives  17000/1 
  So adding to both sides we finally get :
   x²-270x+18225 = 17000

Adding  18225  has completed the left hand side into a perfect square :
   x²-270x+18225  =
   (x-135) • (x-135)  =
  (x-135)²
Things which are equal to the same thing are also equal to one another. Since
   x²-270x+18225 = 17000 and
   x²-270x+18225 = (x-135)²
then, according to the law of transitivity,
   (x-135)² = 17000

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-135)²   is
   (x-135)^2/2 =
  (x-135)¹ =
   x-135

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-135 = √ 17000

Add  135  to both sides to obtain:
   x = 135 + √ 17000

Since a square root has two values, one positive and the other negative
   x² - 270x + 1225 = 0
   has two solutions:
  x = 135 + √ 17000
   or
  x = 135 - √ 17000

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    -x²+270x-1225 = 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   =   270
                      C   =  -1225

Accordingly,  B²  -  4AC   =
                     72900 - 4900 =
                     68000

Applying the quadratic formula :

               -270 ± √ 68000
   x  =    ————————
                        -2

Can  √ 68000 be simplified ?

Yes!   The prime factorization of  68000   is
   2•2•2•2•2•5•5•5•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).

√ 68000   =  √ 2•2•2•2•2•5•5•5•17   =2•2•5•√ 170   =
                ±  20 • √ 170

  √ 170   , rounded to 4 decimal digits, is  13.0384
 So now we are looking at:
           x  =  ( -270 ± 20 •  13.038 ) / -2

Two real solutions:

 x =(-270+√68000)/-2=135-10√ 170 = 4.616

or:

 x =(-270-√68000)/-2=135+10√ 170 = 265.384

Two solutions were found :

1.  x =(-270-√68000)/-2=135+10√ 170 = 265.384
2.  x =(-270+√68000)/-2=135-10√ 170 = 4.616