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  :

  (3x2 -  8x) -  1280  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  3x²-8x-1280 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  -8x  its coefficient is  -8 .
The last term, "the constant", is  -1280 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -1280 = -3840 

Step-2 : Find two factors of  -3840  whose sum equals the coefficient of the middle term, which is   -8 .

For tidiness, printing of 30 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  :

  3x2 - 8x - 1280  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 3x²-8x-1280

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, 3 , 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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   1.3333  

 Plugging into the parabola formula   1.3333  for  x  we can calculate the  y -coordinate : 
  y = 3.0 * 1.33 * 1.33 - 8.0 * 1.33 - 1280.0
or   y = -1285.333

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 3x²-8x-1280
Axis of Symmetry (dashed)  {x}={ 1.33} 
Vertex at  {x,y} = { 1.33,-1285.33} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-19.37, 0.00} 
Root 2 at  {x,y} = {22.03, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   3x²-8x-1280 = 0 by Completing The Square .

 Divide both sides of the equation by  3  to have 1 as the coefficient of the first term :
   x²-(8/3)x-(1280/3) = 0

Add  1280/3  to both side of the equation :
   x²-(8/3)x = 1280/3

Now the clever bit: Take the coefficient of  x , which is  8/3 , divide by two, giving  4/3 , and finally square it giving  16/9 

Add  16/9  to both sides of the equation :
  On the right hand side we have :
   1280/3  +  16/9   The common denominator of the two fractions is  9   Adding  (3840/9)+(16/9)  gives  3856/9 
  So adding to both sides we finally get :
   x²-(8/3)x+(16/9) = 3856/9

Adding  16/9  has completed the left hand side into a perfect square :
   x²-(8/3)x+(16/9)  =
   (x-(4/3)) • (x-(4/3))  =
  (x-(4/3))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(8/3)x+(16/9) = 3856/9 and
   x²-(8/3)x+(16/9) = (x-(4/3))²
then, according to the law of transitivity,
   (x-(4/3))² = 3856/9

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-(4/3))²   is
   (x-(4/3))^2/2 =
  (x-(4/3))¹ =
   x-(4/3)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(4/3) = √ 3856/9

Add  4/3  to both sides to obtain:
   x = 4/3 + √ 3856/9

Since a square root has two values, one positive and the other negative
   x² - (8/3)x - (1280/3) = 0
   has two solutions:
  x = 4/3 + √ 3856/9
   or
  x = 4/3 - √ 3856/9

Note that  √ 3856/9 can be written as
  √ 3856  / √ 9   which is √ 3856  / 3

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    3x²-8x-1280 = 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   =     3
                      B   =    -8
                      C   =  -1280

Accordingly,  B²  -  4AC   =
                     64 - (-15360) =
                     15424

Applying the quadratic formula :

               8 ± √ 15424
   x  =    ——————
                      6

Can  √ 15424 be simplified ?

Yes!   The prime factorization of  15424   is
   2•2•2•2•2•2•241 
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).

√ 15424   =  √ 2•2•2•2•2•2•241   =2•2•2•√ 241   =
                ±  8 • √ 241

  √ 241   , rounded to 4 decimal digits, is  15.5242
 So now we are looking at:
           x  =  ( 8 ± 8 •  15.524 ) / 6

Two real solutions:

 x =(8+√15424)/6=(4+4√ 241 )/3= 22.032

or:

 x =(8-√15424)/6=(4-4√ 241 )/3= -19.366

Two solutions were found :

1.  x =(8-√15424)/6=(4-4√ 241 )/3= -19.366
2.  x =(8+√15424)/6=(4+4√ 241 )/3= 22.032