Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "432.5" was replaced by "(4325/10)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            865
 Simplify   ———
             2 

Equation at the end of step  1  :

             225     865
  ((6•(x2))-(———•x))-———  = 0 
             10       2 

Step  2  :

            45
 Simplify   ——
            2 

Equation at the end of step  2  :

                  45          865
  ((6 • (x2)) -  (—— • x)) -  ———  = 0 
                  2            2 
 Step  3  :

Equation at the end of step  3  :

              45x     865
  ((2•3x2) -  ———) -  ———  = 0 
               2       2 

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  2  as the denominator :

                (2•3x2)     (2•3x2) • 2
     (2•3x2) =  ———————  =  ———————————
                   1             2     

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (2•3x2) • 2 - (45x)     12x2 - 45x
 ———————————————————  =  ——————————
          2                  2     

Equation at the end of step  4  :

  (12x2 - 45x)    865
  ———————————— -  ———  = 0 
       2           2 

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   12x² - 45x  =   3x • (4x - 15) 

Adding fractions which have a common denominator :

 6.2       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 3x • (4x-15) - (865)     12x2 - 45x - 865
 ————————————————————  =  ————————————————
          2                      2        

Trying to factor by splitting the middle term

 6.3     Factoring  12x² - 45x - 865 

The first term is,  12x²  its coefficient is  12 .
The middle term is,  -45x  its coefficient is  -45 .
The last term, "the constant", is  -865 

Step-1 : Multiply the coefficient of the first term by the constant   12 • -865 = -10380 

Step-2 : Find two factors of  -10380  whose sum equals the coefficient of the middle term, which is   -45 .

For tidiness, printing of 18 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  6  :

  12x2 - 45x - 865
  ————————————————  = 0 
         2        

Step  7  :

When a fraction equals zero :

 7.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  12x2-45x-865
  ———————————— • 2 = 0 • 2
       2      

Now, on the left hand side, the  2  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   12x²-45x-865  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = 12x²-45x-865

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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   1.8750  

 Plugging into the parabola formula   1.8750  for  x  we can calculate the  y -coordinate : 
  y = 12.0 * 1.88 * 1.88 - 45.0 * 1.88 - 865.0
or   y = -907.188

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 12x²-45x-865
Axis of Symmetry (dashed)  {x}={ 1.88} 
Vertex at  {x,y} = { 1.88,-907.19} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-6.82, 0.00} 
Root 2 at  {x,y} = {10.57, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   12x²-45x-865 = 0 by Completing The Square .

 Divide both sides of the equation by  12  to have 1 as the coefficient of the first term :
   x²-(15/4)x-(865/12) = 0

Add  865/12  to both side of the equation :
   x²-(15/4)x = 865/12

Now the clever bit: Take the coefficient of  x , which is  15/4 , divide by two, giving  15/8 , and finally square it giving  225/64 

Add  225/64  to both sides of the equation :
  On the right hand side we have :
   865/12  +  225/64   The common denominator of the two fractions is  192   Adding  (13840/192)+(675/192)  gives  14515/192 
  So adding to both sides we finally get :
   x²-(15/4)x+(225/64) = 14515/192

Adding  225/64  has completed the left hand side into a perfect square :
   x²-(15/4)x+(225/64)  =
   (x-(15/8)) • (x-(15/8))  =
  (x-(15/8))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(15/4)x+(225/64) = 14515/192 and
   x²-(15/4)x+(225/64) = (x-(15/8))²
then, according to the law of transitivity,
   (x-(15/8))² = 14515/192

We'll refer to this Equation as  Eq. #7.3.1  

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

Note that the square root of
   (x-(15/8))²   is
   (x-(15/8))^2/2 =
  (x-(15/8))¹ =
   x-(15/8)

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   x-(15/8) = √ 14515/192

Add  15/8  to both sides to obtain:
   x = 15/8 + √ 14515/192

Since a square root has two values, one positive and the other negative
   x² - (15/4)x - (865/12) = 0
   has two solutions:
  x = 15/8 + √ 14515/192
   or
  x = 15/8 - √ 14515/192

Note that  √ 14515/192 can be written as
  √ 14515  / √ 192 

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    12x²-45x-865 = 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   =     12
                      B   =   -45
                      C   =  -865

Accordingly,  B²  -  4AC   =
                     2025 - (-41520) =
                     43545

Applying the quadratic formula :

               45 ± √ 43545
   x  =    ———————
                        24

  √ 43545   , rounded to 4 decimal digits, is  208.6744
 So now we are looking at:
           x  =  ( 45 ±  208.674 ) / 24

Two real solutions:

 x =(45+√43545)/24=15/8+1/24√ 43545 = 10.570

or:

 x =(45-√43545)/24=15/8-1/24√ 43545 = -6.820

Two solutions were found :

1.  x =(45-√43545)/24=15/8-1/24√ 43545 = -6.820
2.  x =(45+√43545)/24=15/8+1/24√ 43545 = 10.570