Solution - Quadratic equations

Rearrange:

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

                     x^2-3/5*x-(60)=0 

Step by step solution :

Step  1  :

            3
 Simplify   —
            5

Equation at the end of step  1  :

            3          
  ((x2) -  (— • x)) -  60  = 0 
            5          
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  5  as the denominator :

           x2     x2 • 5
     x2 =  ——  =  ——————
           1        5   

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 :

 2.2       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:

 x2 • 5 - (3x)     5x2 - 3x
 —————————————  =  ————————
       5              5    

Equation at the end of step  2  :

  (5x2 - 3x)    
  —————————— -  60  = 0 
      5         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5  as the denominator :

          60     60 • 5
    60 =  ——  =  ——————
          1        5   

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   5x² - 3x  =   x • (5x - 3) 

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions

 x • (5x-3) - (60 • 5)     5x2 - 3x - 300
 —————————————————————  =  ——————————————
           5                     5       

Trying to factor by splitting the middle term

 4.3     Factoring  5x² - 3x - 300 

The first term is,  5x²  its coefficient is  5 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  -300 

Step-1 : Multiply the coefficient of the first term by the constant   5 • -300 = -1500 

Step-2 : Find two factors of  -1500  whose sum equals the coefficient of the middle term, which is   -3 .

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  4  :

  5x2 - 3x - 300
  ——————————————  = 0 
        5       

Step  5  :

When a fraction equals zero :

 5.1    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:

  5x2-3x-300
  —————————— • 5 = 0 • 5
      5     

Now, on the left hand side, the  5  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   5x²-3x-300  = 0

Parabola, Finding the Vertex :

 5.2      Find the Vertex of   y = 5x²-3x-300

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, 5 , 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   0.3000  

 Plugging into the parabola formula   0.3000  for  x  we can calculate the  y -coordinate : 
  y = 5.0 * 0.30 * 0.30 - 3.0 * 0.30 - 300.0
or   y = -300.450

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 5x²-3x-300
Axis of Symmetry (dashed)  {x}={ 0.30} 
Vertex at  {x,y} = { 0.30,-300.45} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-7.45, 0.00} 
Root 2 at  {x,y} = { 8.05, 0.00} 

Solve Quadratic Equation by Completing The Square

 5.3     Solving   5x²-3x-300 = 0 by Completing The Square .

 Divide both sides of the equation by  5  to have 1 as the coefficient of the first term :
   x²-(3/5)x-60 = 0

Add  60  to both side of the equation :
   x²-(3/5)x = 60

Now the clever bit: Take the coefficient of  x , which is  3/5 , divide by two, giving  3/10 , and finally square it giving  9/100 

Add  9/100  to both sides of the equation :
  On the right hand side we have :
   60  +  9/100    or,  (60/1)+(9/100) 
  The common denominator of the two fractions is  100   Adding  (6000/100)+(9/100)  gives  6009/100 
  So adding to both sides we finally get :
   x²-(3/5)x+(9/100) = 6009/100

Adding  9/100  has completed the left hand side into a perfect square :
   x²-(3/5)x+(9/100)  =
   (x-(3/10)) • (x-(3/10))  =
  (x-(3/10))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(3/5)x+(9/100) = 6009/100 and
   x²-(3/5)x+(9/100) = (x-(3/10))²
then, according to the law of transitivity,
   (x-(3/10))² = 6009/100

We'll refer to this Equation as  Eq. #5.3.1  

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

Note that the square root of
   (x-(3/10))²   is
   (x-(3/10))^2/2 =
  (x-(3/10))¹ =
   x-(3/10)

Now, applying the Square Root Principle to  Eq. #5.3.1  we get:
   x-(3/10) = √ 6009/100

Add  3/10  to both sides to obtain:
   x = 3/10 + √ 6009/100

Since a square root has two values, one positive and the other negative
   x² - (3/5)x - 60 = 0
   has two solutions:
  x = 3/10 + √ 6009/100
   or
  x = 3/10 - √ 6009/100

Note that  √ 6009/100 can be written as
  √ 6009  / √ 100   which is √ 6009  / 10

Solve Quadratic Equation using the Quadratic Formula

 5.4     Solving    5x²-3x-300 = 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   =     5
                      B   =    -3
                      C   =  -300

Accordingly,  B²  -  4AC   =
                     9 - (-6000) =
                     6009

Applying the quadratic formula :

               3 ± √ 6009
   x  =    ——————
                      10

  √ 6009   , rounded to 4 decimal digits, is  77.5177
 So now we are looking at:
           x  =  ( 3 ±  77.518 ) / 10

Two real solutions:

 x =(3+√6009)/10= 8.052

or:

 x =(3-√6009)/10=-7.452

Two solutions were found :

1.  x =(3-√6009)/10=-7.452
2.  x =(3+√6009)/10= 8.052