Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((24•3x2) +  30x) -  27  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   48x² + 30x - 27  =   3 • (16x² + 10x - 9) 

Trying to factor by splitting the middle term

 3.2     Factoring  16x² + 10x - 9 

The first term is,  16x²  its coefficient is  16 .
The middle term is,  +10x  its coefficient is  10 .
The last term, "the constant", is  -9 

Step-1 : Multiply the coefficient of the first term by the constant   16 • -9 = -144 

Step-2 : Find two factors of  -144  whose sum equals the coefficient of the middle term, which is   10 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -8  and  18 
                     16x² - 8x + 18x - 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    8x • (2x-1)
              Add up the last 2 terms, pulling out common factors :
                    9 • (2x-1)
Step-5 : Add up the four terms of step 4 :
                    (8x+9)  •  (2x-1)
             Which is the desired factorization

Equation at the end of step  3  :

  3 • (2x - 1) • (8x + 9)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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.

Equations which are never true :

 4.2      Solve :    3   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.3      Solve  :    2x-1 = 0 

 Add  1  to both sides of the equation : 
                      2x = 1
Divide both sides of the equation by 2:
                     x = 1/2 = 0.500

Solving a Single Variable Equation :

 4.4      Solve  :    8x+9 = 0 

 Subtract  9  from both sides of the equation : 
                      8x = -9
Divide both sides of the equation by 8:
                     x = -9/8 = -1.125

Supplement : Solving Quadratic Equation Directly

Solving    16x2+10x-9  = 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 :

 5.1      Find the Vertex of   y = 16x²+10x-9

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, 16 , 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  -0.3125  

 Plugging into the parabola formula  -0.3125  for  x  we can calculate the  y -coordinate : 
  y = 16.0 * -0.31 * -0.31 + 10.0 * -0.31 - 9.0
or   y = -10.562

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 16x²+10x-9
Axis of Symmetry (dashed)  {x}={-0.31} 
Vertex at  {x,y} = {-0.31,-10.56} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.12, 0.00} 
Root 2 at  {x,y} = { 0.50, 0.00} 

Solve Quadratic Equation by Completing The Square

 5.2     Solving   16x²+10x-9 = 0 by Completing The Square .

 Divide both sides of the equation by  16  to have 1 as the coefficient of the first term :
   x²+(5/8)x-(9/16) = 0

Add  9/16  to both side of the equation :
   x²+(5/8)x = 9/16

Now the clever bit: Take the coefficient of  x , which is  5/8 , divide by two, giving  5/16 , and finally square it giving  25/256 

Add  25/256  to both sides of the equation :
  On the right hand side we have :
   9/16  +  25/256   The common denominator of the two fractions is  256   Adding  (144/256)+(25/256)  gives  169/256 
  So adding to both sides we finally get :
   x²+(5/8)x+(25/256) = 169/256

Adding  25/256  has completed the left hand side into a perfect square :
   x²+(5/8)x+(25/256)  =
   (x+(5/16)) • (x+(5/16))  =
  (x+(5/16))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(5/8)x+(25/256) = 169/256 and
   x²+(5/8)x+(25/256) = (x+(5/16))²
then, according to the law of transitivity,
   (x+(5/16))² = 169/256

We'll refer to this Equation as  Eq. #5.2.1  

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

Note that the square root of
   (x+(5/16))²   is
   (x+(5/16))^2/2 =
  (x+(5/16))¹ =
   x+(5/16)

Now, applying the Square Root Principle to  Eq. #5.2.1  we get:
   x+(5/16) = √ 169/256

Subtract  5/16  from both sides to obtain:
   x = -5/16 + √ 169/256

Since a square root has two values, one positive and the other negative
   x² + (5/8)x - (9/16) = 0
   has two solutions:
  x = -5/16 + √ 169/256
   or
  x = -5/16 - √ 169/256

Note that  √ 169/256 can be written as
  √ 169  / √ 256   which is 13 / 16

Solve Quadratic Equation using the Quadratic Formula

 5.3     Solving    16x²+10x-9 = 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   =     16
                      B   =    10
                      C   =   -9

Accordingly,  B²  -  4AC   =
                     100 - (-576) =
                     676

Applying the quadratic formula :

               -10 ± √ 676
   x  =    ——————
                      32

Can  √ 676 be simplified ?

Yes!   The prime factorization of  676   is
   2•2•13•13 
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).

√ 676   =  √ 2•2•13•13   =2•13•√ 1   =
                ±  26 • √ 1   =
                ±  26

So now we are looking at:
           x  =  ( -10 ± 26) / 32

Two real solutions:

x =(-10+√676)/32=(-5+13)/16= 0.500

or:

x =(-10-√676)/32=(-5-13)/16= -1.125

Two solutions were found :

1.  x = -9/8 = -1.125
   
2.  x = 1/2 = 0.500