Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  s²+70s-600 

The first term is,  s²  its coefficient is  1 .
The middle term is,  +70s  its coefficient is  70 .
The last term, "the constant", is  -600 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -600 = -600 

Step-2 : Find two factors of  -600  whose sum equals the coefficient of the middle term, which is   70 .

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  1  :

  s2 + 70s - 600  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = s²+70s-600

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,As²+Bs+C,the  s -coordinate of the vertex is given by  -B/(2A) . In our case the  s  coordinate is  -35.0000  

 Plugging into the parabola formula  -35.0000  for  s  we can calculate the  y -coordinate : 
  y = 1.0 * -35.00 * -35.00 + 70.0 * -35.00 - 600.0
or   y = -1825.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = s²+70s-600
Axis of Symmetry (dashed)  {s}={-35.00} 
Vertex at  {s,y} = {-35.00,-1825.00} 
 s -Intercepts (Roots) :
Root 1 at  {s,y} = {-77.72, 0.00} 
Root 2 at  {s,y} = { 7.72, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   s²+70s-600 = 0 by Completing The Square .

 Add  600  to both side of the equation :
   s²+70s = 600

Now the clever bit: Take the coefficient of  s , which is  70 , divide by two, giving  35 , and finally square it giving  1225 

Add  1225  to both sides of the equation :
  On the right hand side we have :
   600  +  1225    or,  (600/1)+(1225/1) 
  The common denominator of the two fractions is  1   Adding  (600/1)+(1225/1)  gives  1825/1 
  So adding to both sides we finally get :
   s²+70s+1225 = 1825

Adding  1225  has completed the left hand side into a perfect square :
   s²+70s+1225  =
   (s+35) • (s+35)  =
  (s+35)²
Things which are equal to the same thing are also equal to one another. Since
   s²+70s+1225 = 1825 and
   s²+70s+1225 = (s+35)²
then, according to the law of transitivity,
   (s+35)² = 1825

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
   (s+35)²   is
   (s+35)^2/2 =
  (s+35)¹ =
   s+35

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   s+35 = √ 1825

Subtract  35  from both sides to obtain:
   s = -35 + √ 1825

Since a square root has two values, one positive and the other negative
   s² + 70s - 600 = 0
   has two solutions:
  s = -35 + √ 1825
   or
  s = -35 - √ 1825

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    s²+70s-600 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  s  , the solution for   As²+Bs+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  s =   ————————
                      2A

  In our case,  A   =     1
                      B   =    70
                      C   =  -600

Accordingly,  B²  -  4AC   =
                     4900 - (-2400) =
                     7300

Applying the quadratic formula :

               -70 ± √ 7300
   s  =    ———————
                        2

Can  √ 7300 be simplified ?

Yes!   The prime factorization of  7300   is
   2•2•5•5•73 
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).

√ 7300   =  √ 2•2•5•5•73   =2•5•√ 73   =
                ±  10 • √ 73

  √ 73   , rounded to 4 decimal digits, is   8.5440
 So now we are looking at:
           s  =  ( -70 ± 10 •  8.544 ) / 2

Two real solutions:

 s =(-70+√7300)/2=-35+5√ 73 = 7.720

or:

 s =(-70-√7300)/2=-35-5√ 73 = -77.720

Two solutions were found :

1.  s =(-70-√7300)/2=-35-5√ 73 = -77.720
2.  s =(-70+√7300)/2=-35+5√ 73 = 7.720