Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "235.2" was replaced by "(2352/10)". 2 more similar replacement(s)

Step by step solution :

Step  1  :

            1176
 Simplify   ————
             5  

Equation at the end of step  1  :

    49        1176
  ((——•(t2))+(————•t))+100  = 0 
    10         5  
 Step  2  :
            49
 Simplify   ——
            10

Equation at the end of step  2  :

    49          1176t     
  ((—— • t2) +  —————) +  100  = 0 
    10            5       

Step  3  :

Equation at the end of step  3  :

   49t2    1176t     
  (———— +  —————) +  100  = 0 
    10       5       

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       10 

      The right denominator is :       5 

      Least Common Multiple:
      10 

Calculating Multipliers :

 4.2    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 = 1

   Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

 4.3      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.      49t2
   ——————————————————  =   ————
         L.C.M              10 

   R. Mult. • R. Num.      1176t • 2
   ——————————————————  =   —————————
         L.C.M                10    

Adding fractions that have a common denominator :

 4.4       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:

 49t2 + 1176t • 2     49t2 + 2352t
 ————————————————  =  ————————————
        10                 10     

Equation at the end of step  4  :

  (49t2 + 2352t)    
  —————————————— +  100  = 0 
        10          

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  10  as the denominator :

           100     100 • 10
    100 =  ———  =  ————————
            1         10   

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

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   49t² + 2352t  =   49t • (t + 48) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 49t • (t+48) + 100 • 10     49t2 + 2352t + 1000
 ———————————————————————  =  ———————————————————
           10                        10         

Trying to factor by splitting the middle term

 6.3     Factoring  49t² + 2352t + 1000 

The first term is,  49t²  its coefficient is  49 .
The middle term is,  +2352t  its coefficient is  2352 .
The last term, "the constant", is  +1000 

Step-1 : Multiply the coefficient of the first term by the constant   49 • 1000 = 49000 

Step-2 : Find two factors of  49000  whose sum equals the coefficient of the middle term, which is   2352 .

For tidiness, printing of 90 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  :

  49t2 + 2352t + 1000
  ———————————————————  = 0 
          10         

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:

  49t2+2352t+1000
  ——————————————— • 10 = 0 • 10
        10       

Now, on the left hand side, the  10  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   49t²+2352t+1000  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = 49t²+2352t+1000

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, 49 , 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,At²+Bt+C,the  t -coordinate of the vertex is given by  -B/(2A) . In our case the  t  coordinate is  -24.0000  

 Plugging into the parabola formula  -24.0000  for  t  we can calculate the  y -coordinate : 
  y = 49.0 * -24.00 * -24.00 + 2352.0 * -24.00 + 1000.0
or   y = -27224.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 49t²+2352t+1000
Axis of Symmetry (dashed)  {t}={-24.00} 
Vertex at  {t,y} = {-24.00,-27224.00} 
 t -Intercepts (Roots) :
Root 1 at  {t,y} = {-47.57, 0.00} 
Root 2 at  {t,y} = {-0.43, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   49t²+2352t+1000 = 0 by Completing The Square .

 Divide both sides of the equation by  49  to have 1 as the coefficient of the first term :
   t²+48t+(1000/49) = 0

Subtract  1000/49  from both side of the equation :
   t²+48t = -1000/49

Now the clever bit: Take the coefficient of  t , which is  48 , divide by two, giving  24 , and finally square it giving  576 

Add  576  to both sides of the equation :
  On the right hand side we have :
   -1000/49  +  576    or,  (-1000/49)+(576/1) 
  The common denominator of the two fractions is  49   Adding  (-1000/49)+(28224/49)  gives  27224/49 
  So adding to both sides we finally get :
   t²+48t+576 = 27224/49

Adding  576  has completed the left hand side into a perfect square :
   t²+48t+576  =
   (t+24) • (t+24)  =
  (t+24)²
Things which are equal to the same thing are also equal to one another. Since
   t²+48t+576 = 27224/49 and
   t²+48t+576 = (t+24)²
then, according to the law of transitivity,
   (t+24)² = 27224/49

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
   (t+24)²   is
   (t+24)^2/2 =
  (t+24)¹ =
   t+24

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   t+24 = √ 27224/49

Subtract  24  from both sides to obtain:
   t = -24 + √ 27224/49

Since a square root has two values, one positive and the other negative
   t² + 48t + (1000/49) = 0
   has two solutions:
  t = -24 + √ 27224/49
   or
  t = -24 - √ 27224/49

Note that  √ 27224/49 can be written as
  √ 27224  / √ 49   which is √ 27224  / 7

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    49t²+2352t+1000 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  t  , the solution for   At²+Bt+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  t =   ————————
                      2A

  In our case,  A   =     49
                      B   =   2352
                      C   =  1000

Accordingly,  B²  -  4AC   =
                     5531904 - 196000 =
                     5335904

Applying the quadratic formula :

               -2352 ± √ 5335904
   t  =    ——————————
                           98

Can  √ 5335904 be simplified ?

Yes!   The prime factorization of  5335904   is
   2•2•2•2•2•7•7•41•83 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 5335904   =  √ 2•2•2•2•2•7•7•41•83   =2•2•7•√ 6806   =
                ±  28 • √ 6806

  √ 6806   , rounded to 4 decimal digits, is  82.4985
 So now we are looking at:
           t  =  ( -2352 ± 28 •  82.498 ) / 98

Two real solutions:

 t =(-2352+√5335904)/98=-24+2/7√ 6806 = -0.429

or:

 t =(-2352-√5335904)/98=-24-2/7√ 6806 = -47.571

Two solutions were found :

1.  t =(-2352-√5335904)/98=-24-2/7√ 6806 = -47.571
2.  t =(-2352+√5335904)/98=-24+2/7√ 6806 = -0.429