Solution - Quadratic equations

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     0-(4*k^2+k-180)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  0 -  ((22k2 +  k) -  180)  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -4k² - k + 180  =   -1 • (4k² + k - 180) 

Trying to factor by splitting the middle term

 3.2     Factoring  4k² + k - 180 

The first term is,  4k²  its coefficient is  4 .
The middle term is,  +k  its coefficient is  1 .
The last term, "the constant", is  -180 

Step-1 : Multiply the coefficient of the first term by the constant   4 • -180 = -720 

Step-2 : Find two factors of  -720  whose sum equals the coefficient of the middle term, which is   1 .

For tidiness, printing of 24 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  :

  -4k2 - k + 180  = 0 

Step  4  :

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = -4k²-k+180

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, -4 , 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,Ak²+Bk+C,the  k -coordinate of the vertex is given by  -B/(2A) . In our case the  k  coordinate is  -0.1250  

 Plugging into the parabola formula  -0.1250  for  k  we can calculate the  y -coordinate : 
  y = -4.0 * -0.12 * -0.12 - 1.0 * -0.12 + 180.0
or   y = 180.062

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -4k²-k+180
Axis of Symmetry (dashed)  {k}={-0.12} 
Vertex at  {k,y} = {-0.12,180.06} 
 k -Intercepts (Roots) :
Root 1 at  {k,y} = { 6.58, 0.00} 
Root 2 at  {k,y} = {-6.83, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   -4k²-k+180 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 4k²+k-180 = 0  Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   k²+(1/4)k-45 = 0

Add  45  to both side of the equation :
   k²+(1/4)k = 45

Now the clever bit: Take the coefficient of  k , which is  1/4 , divide by two, giving  1/8 , and finally square it giving  1/64 

Add  1/64  to both sides of the equation :
  On the right hand side we have :
   45  +  1/64    or,  (45/1)+(1/64) 
  The common denominator of the two fractions is  64   Adding  (2880/64)+(1/64)  gives  2881/64 
  So adding to both sides we finally get :
   k²+(1/4)k+(1/64) = 2881/64

Adding  1/64  has completed the left hand side into a perfect square :
   k²+(1/4)k+(1/64)  =
   (k+(1/8)) • (k+(1/8))  =
  (k+(1/8))²
Things which are equal to the same thing are also equal to one another. Since
   k²+(1/4)k+(1/64) = 2881/64 and
   k²+(1/4)k+(1/64) = (k+(1/8))²
then, according to the law of transitivity,
   (k+(1/8))² = 2881/64

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
   (k+(1/8))²   is
   (k+(1/8))^2/2 =
  (k+(1/8))¹ =
   k+(1/8)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   k+(1/8) = √ 2881/64

Subtract  1/8  from both sides to obtain:
   k = -1/8 + √ 2881/64

Since a square root has two values, one positive and the other negative
   k² + (1/4)k - 45 = 0
   has two solutions:
  k = -1/8 + √ 2881/64
   or
  k = -1/8 - √ 2881/64

Note that  √ 2881/64 can be written as
  √ 2881  / √ 64   which is √ 2881  / 8

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    -4k²-k+180 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  k  , the solution for   Ak²+Bk+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  k =   ————————
                      2A

  In our case,  A   =     -4
                      B   =    -1
                      C   =  180

Accordingly,  B²  -  4AC   =
                     1 - (-2880) =
                     2881

Applying the quadratic formula :

               1 ± √ 2881
   k  =    ——————
                      -8

  √ 2881   , rounded to 4 decimal digits, is  53.6749
 So now we are looking at:
           k  =  ( 1 ±  53.675 ) / -8

Two real solutions:

 k =(1+√2881)/-8=-6.834

or:

 k =(1-√2881)/-8= 6.584

Two solutions were found :

1.  k =(1-√2881)/-8= 6.584
2.  k =(1+√2881)/-8=-6.834