Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((0 -  (2•137x2)) -  617x) +  702  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -274x² - 617x + 702  =   -1 • (274x² + 617x - 702) 

Trying to factor by splitting the middle term

 3.2     Factoring  274x² + 617x - 702 

The first term is,  274x²  its coefficient is  274 .
The middle term is,  +617x  its coefficient is  617 .
The last term, "the constant", is  -702 

Step-1 : Multiply the coefficient of the first term by the constant   274 • -702 = -192348 

Step-2 : Find two factors of  -192348  whose sum equals the coefficient of the middle term, which is   617 .

For tidiness, printing of 42 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  3  :

  -274x2 - 617x + 702  = 0 

Step  4  :

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = -274x²-617x+702

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, -274 , 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  -1.1259  

 Plugging into the parabola formula  -1.1259  for  x  we can calculate the  y -coordinate : 
  y = -274.0 * -1.13 * -1.13 - 617.0 * -1.13 + 702.0
or   y = 1049.344

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -274x²-617x+702
Axis of Symmetry (dashed)  {x}={-1.13} 
Vertex at  {x,y} = {-1.13,1049.34} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.83, 0.00} 
Root 2 at  {x,y} = {-3.08, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   -274x²-617x+702 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 274x²+617x-702 = 0  Divide both sides of the equation by  274  to have 1 as the coefficient of the first term :
   x²+(617/274)x-(351/137) = 0

Add  351/137  to both side of the equation :
   x²+(617/274)x = 351/137

Now the clever bit: Take the coefficient of  x , which is  617/274 , divide by two, giving  617/548 , and finally square it giving  617/548 

Add  617/548  to both sides of the equation :
  On the right hand side we have :
   351/137  +  617/548   The common denominator of the two fractions is  548   Adding  (1404/548)+(617/548)  gives  2021/548 
  So adding to both sides we finally get :
   x²+(617/274)x+(617/548) = 2021/548

Adding  617/548  has completed the left hand side into a perfect square :
   x²+(617/274)x+(617/548)  =
   (x+(617/548)) • (x+(617/548))  =
  (x+(617/548))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(617/274)x+(617/548) = 2021/548 and
   x²+(617/274)x+(617/548) = (x+(617/548))²
then, according to the law of transitivity,
   (x+(617/548))² = 2021/548

We'll refer to this Equation as  Eq. #4.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x+(617/548))²   is
   (x+(617/548))^2/2 =
  (x+(617/548))¹ =
   x+(617/548)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x+(617/548) = √ 2021/548

Subtract  617/548  from both sides to obtain:
   x = -617/548 + √ 2021/548

Since a square root has two values, one positive and the other negative
   x² + (617/274)x - (351/137) = 0
   has two solutions:
  x = -617/548 + √ 2021/548
   or
  x = -617/548 - √ 2021/548

Note that  √ 2021/548 can be written as
  √ 2021  / √ 548 

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    -274x²-617x+702 = 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   =    -274
                      B   =   -617
                      C   =  702

Accordingly,  B²  -  4AC   =
                     380689 - (-769392) =
                     1150081

Applying the quadratic formula :

               617 ± √ 1150081
   x  =    —————————
                          -548

  √ 1150081   , rounded to 4 decimal digits, is  1072.4183
 So now we are looking at:
           x  =  ( 617 ±  1072.418 ) / -548

Two real solutions:

 x =(617+√1150081)/-548=-3.083

or:

 x =(617-√1150081)/-548= 0.831

Two solutions were found :

1.  x =(617-√1150081)/-548= 0.831
2.  x =(617+√1150081)/-548=-3.083