Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22x2 -  115x) +  375  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  4x²-115x+375 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  -115x  its coefficient is  -115 .
The last term, "the constant", is  +375 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 375 = 1500 

Step-2 : Find two factors of  1500  whose sum equals the coefficient of the middle term, which is   -115 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -100  and  -15 
                     4x² - 100x - 15x - 375

Step-4 : Add up the first 2 terms, pulling out like factors :
                    4x • (x-25)
              Add up the last 2 terms, pulling out common factors :
                    15 • (x-25)
Step-5 : Add up the four terms of step 4 :
                    (4x-15)  •  (x-25)
             Which is the desired factorization

Equation at the end of step  2  :

  (x - 25) • (4x - 15)  = 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  :    x-25 = 0 

 Add  25  to both sides of the equation : 
                      x = 25

Solving a Single Variable Equation :

 3.3      Solve  :    4x-15 = 0 

 Add  15  to both sides of the equation : 
                      4x = 15
Divide both sides of the equation by 4:
                     x = 15/4 = 3.750

Supplement : Solving Quadratic Equation Directly

Solving    4x2-115x+375  = 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 = 4x²-115x+375

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, 4 , 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  14.3750  

 Plugging into the parabola formula  14.3750  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 14.38 * 14.38 - 115.0 * 14.38 + 375.0
or   y = -451.562

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x²-115x+375
Axis of Symmetry (dashed)  {x}={14.38} 
Vertex at  {x,y} = {14.38,-451.56} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 3.75, 0.00} 
Root 2 at  {x,y} = {25.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   4x²-115x+375 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x²-(115/4)x+(375/4) = 0

Subtract  375/4  from both side of the equation :
   x²-(115/4)x = -375/4

Now the clever bit: Take the coefficient of  x , which is  115/4 , divide by two, giving  115/8 , and finally square it giving  13225/64 

Add  13225/64  to both sides of the equation :
  On the right hand side we have :
   -375/4  +  13225/64   The common denominator of the two fractions is  64   Adding  (-6000/64)+(13225/64)  gives  7225/64 
  So adding to both sides we finally get :
   x²-(115/4)x+(13225/64) = 7225/64

Adding  13225/64  has completed the left hand side into a perfect square :
   x²-(115/4)x+(13225/64)  =
   (x-(115/8)) • (x-(115/8))  =
  (x-(115/8))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(115/4)x+(13225/64) = 7225/64 and
   x²-(115/4)x+(13225/64) = (x-(115/8))²
then, according to the law of transitivity,
   (x-(115/8))² = 7225/64

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-(115/8))²   is
   (x-(115/8))^2/2 =
  (x-(115/8))¹ =
   x-(115/8)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x-(115/8) = √ 7225/64

Add  115/8  to both sides to obtain:
   x = 115/8 + √ 7225/64

Since a square root has two values, one positive and the other negative
   x² - (115/4)x + (375/4) = 0
   has two solutions:
  x = 115/8 + √ 7225/64
   or
  x = 115/8 - √ 7225/64

Note that  √ 7225/64 can be written as
  √ 7225  / √ 64   which is 85 / 8

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    4x²-115x+375 = 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   =     4
                      B   =   -115
                      C   =  375

Accordingly,  B²  -  4AC   =
                     13225 - 6000 =
                     7225

Applying the quadratic formula :

               115 ± √ 7225
   x  =    ———————
                        8

Can  √ 7225 be simplified ?

Yes!   The prime factorization of  7225   is
   5•5•17•17 
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).

√ 7225   =  √ 5•5•17•17   =5•17•√ 1   =
                ±  85 • √ 1   =
                ±  85

So now we are looking at:
           x  =  ( 115 ± 85) / 8

Two real solutions:

x =(115+√7225)/8=(115+85)/8= 25.000

or:

x =(115-√7225)/8=(115-85)/8= 3.750

Two solutions were found :

1.  x = 15/4 = 3.750
   
2.  x = 25