Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.1" was replaced by "(1/10)". 2 more similar replacement(s)

Rearrange:

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

                4*x^2-((14/100)*((1/10)-x))=0 

Step by step solution :

Step  1  :

             1
 Simplify   ——
            10

Equation at the end of step  1  :

             14   1
  (4•(x2))-(———•(——-x))  = 0 
            100  10

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

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

         x     x • 10
    x =  —  =  ——————
         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

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:

 1 - (x • 10)     1 - 10x
 ————————————  =  ———————
      10            10   

Equation at the end of step  2  :

                  14   (1 - 10x)
  (4 • (x2)) -  (——— • —————————)  = 0 
                 100      10    

Step  3  :

             7
 Simplify   ——
            50

Equation at the end of step  3  :

                  7   (1 - 10x)
  (4 • (x2)) -  (—— • —————————)  = 0 
                 50      10    

Step  4  :

Equation at the end of step  4  :

                7 • (1 - 10x)
  (4 • (x2)) -  —————————————  = 0 
                     500     
 Step  5  :

Equation at the end of step  5  :

          7 • (1 - 10x)
  22x2 -  —————————————  = 0 
               500     

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  500  as the denominator :

             22x2     22x2 • 500
     22x2 =  ————  =  ——————————
              1          500    

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 22x2 • 500 - (7 • (1-10x))     2000x2 + 70x - 7
 ——————————————————————————  =  ————————————————
            500                       500       

Trying to factor by splitting the middle term

 6.3     Factoring  2000x² + 70x - 7 

The first term is,  2000x²  its coefficient is  2000 .
The middle term is,  +70x  its coefficient is  70 .
The last term, "the constant", is  -7 

Step-1 : Multiply the coefficient of the first term by the constant   2000 • -7 = -14000 

Step-2 : Find two factors of  -14000  whose sum equals the coefficient of the middle term, which is   70 .

For tidiness, printing of 34 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  :

  2000x2 + 70x - 7
  ————————————————  = 0 
        500       

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:

  2000x2+70x-7
  ———————————— • 500 = 0 • 500
      500     

Now, on the left hand side, the  500  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   2000x²+70x-7  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = 2000x²+70x-7

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, 2000 , 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  -0.0175  

 Plugging into the parabola formula  -0.0175  for  x  we can calculate the  y -coordinate : 
  y = 2000.0 * -0.02 * -0.02 + 70.0 * -0.02 - 7.0
or   y = -7.612

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 2000x²+70x-7
Axis of Symmetry (dashed)  {x}={-0.02} 
Vertex at  {x,y} = {-0.02,-7.61} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.08, 0.00} 
Root 2 at  {x,y} = { 0.04, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   2000x²+70x-7 = 0 by Completing The Square .

 Divide both sides of the equation by  2000  to have 1 as the coefficient of the first term :
   x²+(7/200)x-(7/2000) = 0

Add  7/2000  to both side of the equation :
   x²+(7/200)x = 7/2000

Now the clever bit: Take the coefficient of  x , which is  7/200 , divide by two, giving  7/400 , and finally square it giving  49/160000 

Add  49/160000  to both sides of the equation :
  On the right hand side we have :
   7/2000  +  49/160000   The common denominator of the two fractions is  160000   Adding  (560/160000)+(49/160000)  gives  609/160000 
  So adding to both sides we finally get :
   x²+(7/200)x+(49/160000) = 609/160000

Adding  49/160000  has completed the left hand side into a perfect square :
   x²+(7/200)x+(49/160000)  =
   (x+(7/400)) • (x+(7/400))  =
  (x+(7/400))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(7/200)x+(49/160000) = 609/160000 and
   x²+(7/200)x+(49/160000) = (x+(7/400))²
then, according to the law of transitivity,
   (x+(7/400))² = 609/160000

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+(7/400))²   is
   (x+(7/400))^2/2 =
  (x+(7/400))¹ =
   x+(7/400)

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   x+(7/400) = √ 609/160000

Subtract  7/400  from both sides to obtain:
   x = -7/400 + √ 609/160000

Since a square root has two values, one positive and the other negative
   x² + (7/200)x - (7/2000) = 0
   has two solutions:
  x = -7/400 + √ 609/160000
   or
  x = -7/400 - √ 609/160000

Note that  √ 609/160000 can be written as
  √ 609  / √ 160000   which is √ 609  / 400

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    2000x²+70x-7 = 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   =    2000
                      B   =    70
                      C   =   -7

Accordingly,  B²  -  4AC   =
                     4900 - (-56000) =
                     60900

Applying the quadratic formula :

               -70 ± √ 60900
   x  =    ————————
                        4000

Can  √ 60900 be simplified ?

Yes!   The prime factorization of  60900   is
   2•2•3•5•5•7•29 
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).

√ 60900   =  √ 2•2•3•5•5•7•29   =2•5•√ 609   =
                ±  10 • √ 609

  √ 609   , rounded to 4 decimal digits, is  24.6779
 So now we are looking at:
           x  =  ( -70 ± 10 •  24.678 ) / 4000

Two real solutions:

 x =(-70+√60900)/4000=(-7+√ 609 )/400= 0.044

or:

 x =(-70-√60900)/4000=(-7-√ 609 )/400= -0.079

Two solutions were found :

1.  x =(-70-√60900)/4000=(-7-√ 609 )/400= -0.079
2.  x =(-70+√60900)/4000=(-7+√ 609 )/400= 0.044