Solve this quadratic equation by completing the square.
x^2+4x+15

Root plot for :  y = x2-4x+15
Axis of Symmetry (dashed)  {x}={ 2.00} 
Vertex at  {x,y} = { 2.00,11.00}  
Function has no real roots

 2.2     Solving   x2-4x+15 = 0 by Completing The Square .

 Subtract  15  from both side of the equation :
   x2-4x = -15

Now the clever bit: Take the coefficient of  x , which is  4 , divide by two, giving  2 , and finally square it giving  4 

Add  4  to both sides of the equation :
  On the right hand side we have :
   -15  +  4    or,  (-15/1)+(4/1) 
  The common denominator of the two fractions is  1   Adding  (-15/1)+(4/1)  gives  -11/1 
  So adding to both sides we finally get :
   x2-4x+4 = -11

Adding  4  has completed the left hand side into a perfect square :
   x2-4x+4  =
   (x-2) • (x-2)  =
  (x-2)2 
Things which are equal to the same thing are also equal to one another. Since
   x2-4x+4 = -11 and
   x2-4x+4 = (x-2)2 
then, according to the law of transitivity,
   (x-2)2 = -11

We'll refer to this Equation as  Eq. #2.2.1  

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

Note that the square root of
   (x-2)2   is
   (x-2)2/2 =
  (x-2)1 =
   x-2

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x-2 = √ -11 

Add  2  to both sides to obtain:
   x = 2 + √ -11 
In Math,  i  is called the imaginary unit. It satisfies   i2  =-1. Both   i   and   -i   are the square roots of   -1 

Since a square root has two values, one positive and the other negative
   x2 - 4x + 15 = 0
   has two solutions:
  x = 2 + √ 11 •  i 
   or
  x = 2 - √ 11 •  i 

 2.3     Solving    x2-4x+15 = 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   =     1
                      B   =    -4
                      C   =   15 

Accordingly,  B2  -  4AC   =
                     16 - 60 =
                     -44

Applying the quadratic formula :

               4 ± √ -44 
   x  =    —————
                    2

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -44  = 
                    √ 44 • (-1)  =
                    √ 44  • √ -1   =
                    ±  √ 44  • i

Can  √ 44 be simplified ?

Yes!   The prime factorization of  44   is
   2•2•11  
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).

√ 44   =  √ 2•2•11   =
                ±  2 • √ 11 

  √ 11   , rounded to 4 decimal digits, is   3.3166
 So now we are looking at:
           x  =  ( 4 ± 2 •  3.317 i ) / 2

Two imaginary solutions : 

.

step-by-step explanation:

first, we define the variables:

x: number of years after 1950

f (x): amount of vinyl sold.

then, with the variables defined, we have:

68594 vinyl records were sold in 1958 > f (8) = 68594

91299 vinyl records were sold in 1961 > f (11) = 91299

38720 vinyl records were sold in 1952 > f (2) = 38720

161743 vinyl records were sold in 1967 > f (17) = 161743

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step-by-step explanation: