Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((23•3m2) -  50m) -  24  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   24m² - 50m - 24  =   2 • (12m² - 25m - 12) 

Trying to factor by splitting the middle term

 3.2     Factoring  12m² - 25m - 12 

The first term is,  12m²  its coefficient is  12 .
The middle term is,  -25m  its coefficient is  -25 .
The last term, "the constant", is  -12 

Step-1 : Multiply the coefficient of the first term by the constant   12 • -12 = -144 

Step-2 : Find two factors of  -144  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  :

  2 • (12m2 - 25m - 12)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 12m²-25m-12

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, 12 , 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,Am²+Bm+C,the  m -coordinate of the vertex is given by  -B/(2A) . In our case the  m  coordinate is   1.0417  

 Plugging into the parabola formula   1.0417  for  m  we can calculate the  y -coordinate : 
  y = 12.0 * 1.04 * 1.04 - 25.0 * 1.04 - 12.0
or   y = -25.021

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 12m²-25m-12
Axis of Symmetry (dashed)  {m}={ 1.04} 
Vertex at  {m,y} = { 1.04,-25.02} 
 m -Intercepts (Roots) :
Root 1 at  {m,y} = {-0.40, 0.00} 
Root 2 at  {m,y} = { 2.49, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.3     Solving   12m²-25m-12 = 0 by Completing The Square .

 Divide both sides of the equation by  12  to have 1 as the coefficient of the first term :
   m²-(25/12)m-1 = 0

Add  1  to both side of the equation :
   m²-(25/12)m = 1

Now the clever bit: Take the coefficient of  m , which is  25/12 , divide by two, giving  25/24 , and finally square it giving  625/576 

Add  625/576  to both sides of the equation :
  On the right hand side we have :
   1  +  625/576    or,  (1/1)+(625/576) 
  The common denominator of the two fractions is  576   Adding  (576/576)+(625/576)  gives  1201/576 
  So adding to both sides we finally get :
   m²-(25/12)m+(625/576) = 1201/576

Adding  625/576  has completed the left hand side into a perfect square :
   m²-(25/12)m+(625/576)  =
   (m-(25/24)) • (m-(25/24))  =
  (m-(25/24))²
Things which are equal to the same thing are also equal to one another. Since
   m²-(25/12)m+(625/576) = 1201/576 and
   m²-(25/12)m+(625/576) = (m-(25/24))²
then, according to the law of transitivity,
   (m-(25/24))² = 1201/576

We'll refer to this Equation as  Eq. #4.3.1  

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

Note that the square root of
   (m-(25/24))²   is
   (m-(25/24))^2/2 =
  (m-(25/24))¹ =
   m-(25/24)

Now, applying the Square Root Principle to  Eq. #4.3.1  we get:
   m-(25/24) = √ 1201/576

Add  25/24  to both sides to obtain:
   m = 25/24 + √ 1201/576

Since a square root has two values, one positive and the other negative
   m² - (25/12)m - 1 = 0
   has two solutions:
  m = 25/24 + √ 1201/576
   or
  m = 25/24 - √ 1201/576

Note that  √ 1201/576 can be written as
  √ 1201  / √ 576   which is √ 1201  / 24

Solve Quadratic Equation using the Quadratic Formula

 4.4     Solving    12m²-25m-12 = 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   =     12
                      B   =   -25
                      C   =  -12

Accordingly,  B²  -  4AC   =
                     625 - (-576) =
                     1201

Applying the quadratic formula :

               25 ± √ 1201
   m  =    ——————
                      24

  √ 1201   , rounded to 4 decimal digits, is  34.6554
 So now we are looking at:
           m  =  ( 25 ±  34.655 ) / 24

Two real solutions:

 m =(25+√1201)/24= 2.486

or:

 m =(25-√1201)/24=-0.402

Two solutions were found :

1.  m =(25-√1201)/24=-0.402
2.  m =(25+√1201)/24= 2.486