Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (212x2 +  96x) -  1  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  4096x²+96x-1 

The first term is,  4096x²  its coefficient is  4096 .
The middle term is,  +96x  its coefficient is  96 .
The last term, "the constant", is  -1 

Step-1 : Multiply the coefficient of the first term by the constant   4096 • -1 = -4096 

Step-2 : Find two factors of  -4096  whose sum equals the coefficient of the middle term, which is   96 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -32  and  128 
                     4096x² - 32x + 128x - 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    32x • (128x-1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (128x-1)
Step-5 : Add up the four terms of step 4 :
                    (32x+1)  •  (128x-1)
             Which is the desired factorization

Equation at the end of step  2  :

  (128x - 1) • (32x + 1)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 3.2      Solve  :    128x-1 = 0 

 Add  1  to both sides of the equation : 
                      128x = 1
Divide both sides of the equation by 128:
                     x = 1/128 = 0.008

Solving a Single Variable Equation :

 3.3      Solve  :    32x+1 = 0 

 Subtract  1  from both sides of the equation : 
                      32x = -1
Divide both sides of the equation by 32:
                     x = -1/32 = -0.031

Supplement : Solving Quadratic Equation Directly

Solving    4096x2+96x-1  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = 4096x²+96x-1

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, 4096 , 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  -0.0117  

 Plugging into the parabola formula  -0.0117  for  x  we can calculate the  y -coordinate : 
  y = 4096.0 * -0.01 * -0.01 + 96.0 * -0.01 - 1.0
or   y = -1.562

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4096x²+96x-1
Axis of Symmetry (dashed)  {x}={-0.01} 
Vertex at  {x,y} = {-0.01,-1.56} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.03, 0.00} 
Root 2 at  {x,y} = { 0.01, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   4096x²+96x-1 = 0 by Completing The Square .

 Divide both sides of the equation by  4096  to have 1 as the coefficient of the first term :
   x²+(3/128)x-(1/4096) = 0

Add  1/4096  to both side of the equation :
   x²+(3/128)x = 1/4096

Now the clever bit: Take the coefficient of  x , which is  3/128 , divide by two, giving  3/256 , and finally square it giving  9/65536 

Add  9/65536  to both sides of the equation :
  On the right hand side we have :
   1/4096  +  9/65536   The common denominator of the two fractions is  65536   Adding  (16/65536)+(9/65536)  gives  25/65536 
  So adding to both sides we finally get :
   x²+(3/128)x+(9/65536) = 25/65536

Adding  9/65536  has completed the left hand side into a perfect square :
   x²+(3/128)x+(9/65536)  =
   (x+(3/256)) • (x+(3/256))  =
  (x+(3/256))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(3/128)x+(9/65536) = 25/65536 and
   x²+(3/128)x+(9/65536) = (x+(3/256))²
then, according to the law of transitivity,
   (x+(3/256))² = 25/65536

We'll refer to this Equation as  Eq. #4.2.1  

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

Note that the square root of
   (x+(3/256))²   is
   (x+(3/256))^2/2 =
  (x+(3/256))¹ =
   x+(3/256)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x+(3/256) = √ 25/65536

Subtract  3/256  from both sides to obtain:
   x = -3/256 + √ 25/65536

Since a square root has two values, one positive and the other negative
   x² + (3/128)x - (1/4096) = 0
   has two solutions:
  x = -3/256 + √ 25/65536
   or
  x = -3/256 - √ 25/65536

Note that  √ 25/65536 can be written as
  √ 25  / √ 65536   which is 5 / 256

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    4096x²+96x-1 = 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   =    4096
                      B   =    96
                      C   =   -1

Accordingly,  B²  -  4AC   =
                     9216 - (-16384) =
                     25600

Applying the quadratic formula :

               -96 ± √ 25600
   x  =    ————————
                        8192

Can  √ 25600 be simplified ?

Yes!   The prime factorization of  25600   is
   2•2•2•2•2•2•2•2•2•2•5•5 
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).

√ 25600   =  √ 2•2•2•2•2•2•2•2•2•2•5•5   =2•2•2•2•2•5•√ 1   =
                ±  160 • √ 1   =
                ±  160

So now we are looking at:
           x  =  ( -96 ± 160) / 8192

Two real solutions:

x =(-96+√25600)/8192=(-3+5)/256= 0.008

or:

x =(-96-√25600)/8192=(-3-5)/256= -0.031

Two solutions were found :

1.  x = -1/32 = -0.031
   
2.  x = 1/128 = 0.008