Please help and explain in detail.

Question

Find the roots of the quadratic equation x^2 – 8x = 9 by completing the square.

Answer 1

There are some quadratic equations that look like Ax² + Bx + C in which A=1, B=2*N and C=N².  In these cases (now, think FOIL),  (X + N)(X + N) are factors (and they are the same, so this expression is the square of this factor) [notice again the values of A, B, and C).

 

Now, given x^2 - 8x -9, we first make sure that A=1   (or divide by coefficient to make it 1)

      Then we find N=B/2 (it looks like it is -4)

      and then find C=N^2   (16)

      so we make that the left side of our equation.

           x^2 - 8x + 16 = 25         (we had to add 16 to both sides)

             (x - 4)(x -4) = 25

                  x - 4 = ±5              (took the square root of both sides)

                   x = -1 or x  = 9               (add 4 to both sides)

 

Checking (very important):

     Is     (-1)^2 - 8(-1) = 9    ?

                  1   + 8      = 9         Yes. (-1) is a root

    Is     (9)^2 -8(9)  = 9   ?

               81  - 72  = 9           Yes. (9) is a root

Answer 2

Suppose we wrote David's solution (which is correct) another way.  Let's start with this.

 

(x - 4)(x - 4) = 25

 

 

We know that (x - 4) is raised to the 2nd power.  This can be written as

 

(x - 4)² = 25

 

 

Square-root both sides of equation,

 

x - 4 = ±√25

 

 

Adding 4 on both sides of the equation to isolate x,

 

x = 4 ± √25

 

 

Now does this form remind you of the quadratic formula?