Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "4.31" was replaced by "(431/100)". 3 more similar replacement(s)

Step by step solution :

Step  1  :

            431
 Simplify   ———
            100

Equation at the end of step  1  :

    432        967     431
  ((———•(x2))-(———•x))+———  = 0 
    100        100     100

Step  2  :

            967
 Simplify   ———
            100

Equation at the end of step  2  :

    432        967     431
  ((———•(x2))-(———•x))+———  = 0 
    100        100     100
 Step  3  :
            108
 Simplify   ———
            25 

Equation at the end of step  3  :

    108          967x     431
  ((——— • x2) -  ————) +  ———  = 0 
    25           100      100

Step  4  :

Equation at the end of step  4  :

   108x2    967x     431
  (————— -  ————) +  ———  = 0 
    25      100      100

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       25 

      The right denominator is :       100 

      Least Common Multiple:
      100 

Calculating Multipliers :

 5.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 4

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      108x2 • 4
   ——————————————————  =   —————————
         L.C.M                100   

   R. Mult. • R. Num.      967x
   ——————————————————  =   ————
         L.C.M             100 

Adding fractions that have a common denominator :

 5.4       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:

 108x2 • 4 - (967x)     432x2 - 967x
 ——————————————————  =  ————————————
        100                 100     

Equation at the end of step  5  :

  (432x2 - 967x)    431
  —————————————— +  ———  = 0 
       100          100

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   432x² - 967x  =   x • (432x - 967) 

Adding fractions which have a common denominator :

 7.2       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:

 x • (432x-967) + 431     432x2 - 967x + 431
 ————————————————————  =  ——————————————————
         100                     100        

Trying to factor by splitting the middle term

 7.3     Factoring  432x² - 967x + 431 

The first term is,  432x²  its coefficient is  432 .
The middle term is,  -967x  its coefficient is  -967 .
The last term, "the constant", is  +431 

Step-1 : Multiply the coefficient of the first term by the constant   432 • 431 = 186192 

Step-2 : Find two factors of  186192  whose sum equals the coefficient of the middle term, which is   -967 .

For tidiness, printing of 74 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  7  :

  432x2 - 967x + 431
  ——————————————————  = 0 
         100        

Step  8  :

When a fraction equals zero :

 8.1    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:

  432x2-967x+431
  —————————————— • 100 = 0 • 100
       100      

Now, on the left hand side, the  100  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   432x²-967x+431  = 0

Parabola, Finding the Vertex :

 8.2      Find the Vertex of   y = 432x²-967x+431

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, 432 , 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   1.1192  

 Plugging into the parabola formula   1.1192  for  x  we can calculate the  y -coordinate : 
  y = 432.0 * 1.12 * 1.12 - 967.0 * 1.12 + 431.0
or   y = -110.139

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 432x²-967x+431
Axis of Symmetry (dashed)  {x}={ 1.12} 
Vertex at  {x,y} = { 1.12,-110.14} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.61, 0.00} 
Root 2 at  {x,y} = { 1.62, 0.00} 

Solve Quadratic Equation by Completing The Square

 8.3     Solving   432x²-967x+431 = 0 by Completing The Square .

 Divide both sides of the equation by  432  to have 1 as the coefficient of the first term :
   x²-(967/432)x+(431/432) = 0

Subtract  431/432  from both side of the equation :
   x²-(967/432)x = -431/432

Now the clever bit: Take the coefficient of  x , which is  967/432 , divide by two, giving  967/864 , and finally square it giving  967/864 

Add  967/864  to both sides of the equation :
  On the right hand side we have :
   -431/432  +  967/864   The common denominator of the two fractions is  864   Adding  (-862/864)+(967/864)  gives  105/864 
  So adding to both sides we finally get :
   x²-(967/432)x+(967/864) = 35/288

Adding  967/864  has completed the left hand side into a perfect square :
   x²-(967/432)x+(967/864)  =
   (x-(967/864)) • (x-(967/864))  =
  (x-(967/864))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(967/432)x+(967/864) = 35/288 and
   x²-(967/432)x+(967/864) = (x-(967/864))²
then, according to the law of transitivity,
   (x-(967/864))² = 35/288

We'll refer to this Equation as  Eq. #8.3.1  

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

Note that the square root of
   (x-(967/864))²   is
   (x-(967/864))^2/2 =
  (x-(967/864))¹ =
   x-(967/864)

Now, applying the Square Root Principle to  Eq. #8.3.1  we get:
   x-(967/864) = √ 35/288

Add  967/864  to both sides to obtain:
   x = 967/864 + √ 35/288

Since a square root has two values, one positive and the other negative
   x² - (967/432)x + (431/432) = 0
   has two solutions:
  x = 967/864 + √ 35/288
   or
  x = 967/864 - √ 35/288

Note that  √ 35/288 can be written as
  √ 35  / √ 288 

Solve Quadratic Equation using the Quadratic Formula

 8.4     Solving    432x²-967x+431 = 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   =    432
                      B   =   -967
                      C   =  431

Accordingly,  B²  -  4AC   =
                     935089 - 744768 =
                     190321

Applying the quadratic formula :

               967 ± √ 190321
   x  =    ————————
                        864

  √ 190321   , rounded to 4 decimal digits, is  436.2580
 So now we are looking at:
           x  =  ( 967 ±  436.258 ) / 864

Two real solutions:

 x =(967+√190321)/864= 1.624

or:

 x =(967-√190321)/864= 0.614

Two solutions were found :

1.  x =(967-√190321)/864= 0.614
2.  x =(967+√190321)/864= 1.624