Quadratic equations

Question

To solve the following quadratic by completing the square, what value would you add to both sides of the equation x² - 6x=7?

Answer 1

Completing the square for a quadratic equation of form:

 

ax^2 + bx +c =0   in general when a=1 is:

 

 

x*2 + bx +(b/2)^2 -(b/2)^2 +c = 0

 

This can also be written as :

 

x^2 +bx +(b/2)^2 = -c + (b/2)^2

 

In the case of    x^2 -6x = 7,

 

Since b =-6, b/2 = -6/2

 

so, (b/2)^2 = (-6/2)^2 = (-3)^2 = 9

 

Which means 9 should be added on both sides.

 

This process is useful when the factoring is not evident as in  this case.

 

Answer 2

Completing the square can be explained using FOIL, so you usually learn FOIL first.

    (x+a)(x+a) = x² + 2ax + a²

    (x+2a)(x+2a) = x² + 4ax + 4a²

    (x+3a)(x+3a) = x² + 6ax + 9a²

    (x-3a)(x-3a) = x² - 6ax + 9a²

So, if you start with                  x² - 6x        = 7

The left side needs to be:          x² - 6x + 9  = 7 + 9

                                         (x-3)(x-3) = 16

Take the square root of both sides:

                                         x - 3 = ± 4

                             either x=7  or  x=-1

 

Checking:

  For x=3,  does   (7)² - 6(7) = 7   ?

                           49 - 42 = 7   ?

                            7 = 7   ?  yes

  For x=-1, does    (-1)² -6(-1) = 7   ?

                             1   +  6    = 7   ?

                             7  =  7   ? yes

Answer 3

Since the coefficient of x² is 1, the number that gets added to both sides is found by taking ½ of the coefficient of x and then squaring the result of that operation.    

 

So, in this particular example, we take half of 6, which is 3, and then square 3 to get 9.  We then add 9 to both sides to obtain:

 

x² - 6x + 9 = 7 + 9  The reason for doing this is that the left side is now a perfect square trinomial.  Hence the name "completing the square".  

 

Thus, (x-3)² = 16

 

         x - 3 = ±√16

 

         x - 3 = ±4

 

             x = 3 + 4 or 3 - 4

 

             x = 7 or -1

 

NOTE:  in the given problem, the coefficient of x² is 1.  If the coefficient of x² isn't 1, we would first divide both sides of the equation by the coefficient of x² and then proceed as above.  

 

For example:  Solve 3x² + 12x = -10 by completing the square

 

 Solution:    Divide both sides by 3 to get x² + 4x = -10/3

 

                 Add (½(4))² = 4 to both sides to obtain x²+4x+4 = 2/3

 

                 Factoring the left side of the equation gives (x+2)² = 2/3

 

               Taking the square root of both sides yields    x+2 = ±√(2/3)

 

               So,  x = -2 ±√(2/3)

 

 

Answer 4

x² - 6x + 9= 7 + 9

(x-3)(x-3) = 16

x-3 = ±4

x = 7, -1

 

this problem can also be solved by subtracting 7 from both sides

x² - 6x -7 = 0

(x-7)(x+1) = 0

x = 7, -1