Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "6.9" was replaced by "(69/10)".

Step by step solution :

Step  1  :

            69
 Simplify   ——
            10

Equation at the end of step  1  :

                         69
  ((3 • (x2)) -  19x) -  ——  = 0 
                         10
 Step  2  :

Equation at the end of step  2  :

                  69
  (3x2 -  19x) -  ——  = 0 
                  10

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

                  3x2 - 19x     (3x2 - 19x) • 10
     3x2 - 19x =  —————————  =  ————————————————
                      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  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   3x² - 19x  =   x • (3x - 19) 

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:

 x • (3x-19) • 10 - (69)     30x2 - 190x - 69
 ———————————————————————  =  ————————————————
           10                       10       

Trying to factor by splitting the middle term

 4.3     Factoring  30x² - 190x - 69 

The first term is,  30x²  its coefficient is  30 .
The middle term is,  -190x  its coefficient is  -190 .
The last term, "the constant", is  -69 

Step-1 : Multiply the coefficient of the first term by the constant   30 • -69 = -2070 

Step-2 : Find two factors of  -2070  whose sum equals the coefficient of the middle term, which is   -190 .

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  :

  30x2 - 190x - 69
  ————————————————  = 0 
         10       

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:

  30x2-190x-69
  ———————————— • 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 :
   30x²-190x-69  = 0

Parabola, Finding the Vertex :

 5.2      Find the Vertex of   y = 30x²-190x-69

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, 30 , 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   3.1667  

 Plugging into the parabola formula   3.1667  for  x  we can calculate the  y -coordinate : 
  y = 30.0 * 3.17 * 3.17 - 190.0 * 3.17 - 69.0
or   y = -369.833

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 30x²-190x-69
Axis of Symmetry (dashed)  {x}={ 3.17} 
Vertex at  {x,y} = { 3.17,-369.83} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.34, 0.00} 
Root 2 at  {x,y} = { 6.68, 0.00} 

Solve Quadratic Equation by Completing The Square

 5.3     Solving   30x²-190x-69 = 0 by Completing The Square .

 Divide both sides of the equation by  30  to have 1 as the coefficient of the first term :
   x²-(19/3)x-(23/10) = 0

Add  23/10  to both side of the equation :
   x²-(19/3)x = 23/10

Now the clever bit: Take the coefficient of  x , which is  19/3 , divide by two, giving  19/6 , and finally square it giving  361/36 

Add  361/36  to both sides of the equation :
  On the right hand side we have :
   23/10  +  361/36   The common denominator of the two fractions is  180   Adding  (414/180)+(1805/180)  gives  2219/180 
  So adding to both sides we finally get :
   x²-(19/3)x+(361/36) = 2219/180

Adding  361/36  has completed the left hand side into a perfect square :
   x²-(19/3)x+(361/36)  =
   (x-(19/6)) • (x-(19/6))  =
  (x-(19/6))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(19/3)x+(361/36) = 2219/180 and
   x²-(19/3)x+(361/36) = (x-(19/6))²
then, according to the law of transitivity,
   (x-(19/6))² = 2219/180

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-(19/6))²   is
   (x-(19/6))^2/2 =
  (x-(19/6))¹ =
   x-(19/6)

Now, applying the Square Root Principle to  Eq. #5.3.1  we get:
   x-(19/6) = √ 2219/180

Add  19/6  to both sides to obtain:
   x = 19/6 + √ 2219/180

Since a square root has two values, one positive and the other negative
   x² - (19/3)x - (23/10) = 0
   has two solutions:
  x = 19/6 + √ 2219/180
   or
  x = 19/6 - √ 2219/180

Note that  √ 2219/180 can be written as
  √ 2219  / √ 180 

Solve Quadratic Equation using the Quadratic Formula

 5.4     Solving    30x²-190x-69 = 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   =     30
                      B   =   -190
                      C   =  -69

Accordingly,  B²  -  4AC   =
                     36100 - (-8280) =
                     44380

Applying the quadratic formula :

               190 ± √ 44380
   x  =    ————————
                        60

Can  √ 44380 be simplified ?

Yes!   The prime factorization of  44380   is
   2•2•5•7•317 
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).

√ 44380   =  √ 2•2•5•7•317   =
                ±  2 • √ 11095

  √ 11095   , rounded to 4 decimal digits, is  105.3328
 So now we are looking at:
           x  =  ( 190 ± 2 •  105.333 ) / 60

Two real solutions:

 x =(190+√44380)/60=19/6+1/30√ 11095 = 6.678

or:

 x =(190-√44380)/60=19/6-1/30√ 11095 = -0.344

Two solutions were found :

1.  x =(190-√44380)/60=19/6-1/30√ 11095 = -0.344
2.  x =(190+√44380)/60=19/6+1/30√ 11095 = 6.678