Solve by completing the square: x^2-9x-38=-9

Question

-Completing the square

- Round final answer to hundredths place

Answer 1

First, let's review what "the square" looks like:
             (x + a)(x + a)
             x^2 + ax + ax + a^2
             x^2 + 2ax + a^2
 
Now, we must find the value for "a" in this expression.
 
The problem gives us x^2 -9x ...     So, 2a=-9   and a=-9/2
 
O.K., that tells us that (x-9/2)(x-9/2) = x^2 -9x + 81/4       [this is a perfect square]
 
Now, what needs to be done to the given equation to make it have that expression on the left side?
    x^2 - 9x -38 = -9
    x^2 - 9x   =  29                 [add 38 to each side]
    x^2 - 9x + 81/4   = 29 +81/4                    [add 81/2 to each side]
    x^2 - 9x + 81/4   = (116+81)/4                [convert to improper fraction]
    x^2 - 9x + 81/4   =  197/4                    [now the left side is O.K.]
    (x-9/2)(x-9/2) = 197/4       
        x-9/2 ≅ 14.04/2    or  x-9/2 ≅ -14.04/2           [take square root of each side]
        x ≅ (14.04 + 9)/2  or   x ≅ (-14.04 + 9)/2            [add 9/2 to each side]
    x ≅ 23.04/2    or     x ≅ -5.04/2

    x ≅ 11.52    or   x≅ -2.52

 

Checking (very important):

  Is   (11.52)^2 - 9(11.52) - 38 ≅ -9   ?

         132.71 - 103.68 - 38 ≅ -9   ?

         -8.97 ≅ -9     ? yes       

 

  Is   (-2.52)^2 - 9(-2,52) -38 ≅ -9    ?

          6.35 + 22.62 - 38 ≅ -9    ?

            -9.03 ≅ -9   ? yes

Answer 2

Add 38 to both sides:   x²-9x = 29

 

Divide -9 by 2, square -9/2, and add that number to both sides:

 

   x² - 9x + 81/4 = 29 + 81/4

 

Factor the left side:  (x - 9/2)² = 197/4

 

Take square roots:  x - 9/2 = ±√(197)/2

 

x = (9 ± √197)/2 ≈11.52, -2.52

 

 

Answer 3

x²-9x-38=-9

=>x2-9x-29=0 => the equation has no rational root.

the complex roots can be ={-9±√(81-(4.1.[-29])}/2

                                       ={-9±(√81+(4.1.29)}/2... where i²=-1 and i is an imaginary number

                                       =  -9/2±√(81+116)/2

                                       =-9/2±(√197)/2

                                       ≈-4.5±7.02

                                        ≈2.52 or -11.52