Solution - Quadratic equations

Rearrange:

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

                     21-(6*m^2+25*m)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  21 -  ((2•3m2) +  25m)  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -6m² - 25m + 21  =   -1 • (6m² + 25m - 21) 

Trying to factor by splitting the middle term

 3.2     Factoring  6m² + 25m - 21 

The first term is,  6m²  its coefficient is  6 .
The middle term is,  +25m  its coefficient is  25 .
The last term, "the constant", is  -21 

Step-1 : Multiply the coefficient of the first term by the constant   6 • -21 = -126 

Step-2 : Find two factors of  -126  whose sum equals the coefficient of the middle term, which is   25 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  3  :

  -6m2 - 25m + 21  = 0 

Step  4  :

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = -6m²-25m+21

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, -6 , 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,Am²+Bm+C,the  m -coordinate of the vertex is given by  -B/(2A) . In our case the  m  coordinate is  -2.0833  

 Plugging into the parabola formula  -2.0833  for  m  we can calculate the  y -coordinate : 
  y = -6.0 * -2.08 * -2.08 - 25.0 * -2.08 + 21.0
or   y = 47.042

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -6m²-25m+21
Axis of Symmetry (dashed)  {m}={-2.08} 
Vertex at  {m,y} = {-2.08,47.04} 
 m -Intercepts (Roots) :
Root 1 at  {m,y} = { 0.72, 0.00} 
Root 2 at  {m,y} = {-4.88, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   -6m²-25m+21 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 6m²+25m-21 = 0  Divide both sides of the equation by  6  to have 1 as the coefficient of the first term :
   m²+(25/6)m-(7/2) = 0

Add  7/2  to both side of the equation :
   m²+(25/6)m = 7/2

Now the clever bit: Take the coefficient of  m , which is  25/6 , divide by two, giving  25/12 , and finally square it giving  625/144 

Add  625/144  to both sides of the equation :
  On the right hand side we have :
   7/2  +  625/144   The common denominator of the two fractions is  144   Adding  (504/144)+(625/144)  gives  1129/144 
  So adding to both sides we finally get :
   m²+(25/6)m+(625/144) = 1129/144

Adding  625/144  has completed the left hand side into a perfect square :
   m²+(25/6)m+(625/144)  =
   (m+(25/12)) • (m+(25/12))  =
  (m+(25/12))²
Things which are equal to the same thing are also equal to one another. Since
   m²+(25/6)m+(625/144) = 1129/144 and
   m²+(25/6)m+(625/144) = (m+(25/12))²
then, according to the law of transitivity,
   (m+(25/12))² = 1129/144

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
   (m+(25/12))²   is
   (m+(25/12))^2/2 =
  (m+(25/12))¹ =
   m+(25/12)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   m+(25/12) = √ 1129/144

Subtract  25/12  from both sides to obtain:
   m = -25/12 + √ 1129/144

Since a square root has two values, one positive and the other negative
   m² + (25/6)m - (7/2) = 0
   has two solutions:
  m = -25/12 + √ 1129/144
   or
  m = -25/12 - √ 1129/144

Note that  √ 1129/144 can be written as
  √ 1129  / √ 144   which is √ 1129  / 12

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    -6m²-25m+21 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  m  , the solution for   Am²+Bm+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  m =   ————————
                      2A

  In our case,  A   =     -6
                      B   =   -25
                      C   =   21

Accordingly,  B²  -  4AC   =
                     625 - (-504) =
                     1129

Applying the quadratic formula :

               25 ± √ 1129
   m  =    ——————
                      -12

  √ 1129   , rounded to 4 decimal digits, is  33.6006
 So now we are looking at:
           m  =  ( 25 ±  33.601 ) / -12

Two real solutions:

 m =(25+√1129)/-12=-4.883

or:

 m =(25-√1129)/-12= 0.717

Two solutions were found :

1.  m =(25-√1129)/-12= 0.717
2.  m =(25+√1129)/-12=-4.883