x² + 18x = -9

Rule for finding the number to add  to both sides to complete the square:

1. Multiply the coefficient of x by 
2. Square the result of step 1.

The coefficient of x is 18

1. 18· = 9
2. 9² = 81

So we must add 81 to both sides

That's all you were asked for.

To solve the equation,

x² + 18x = -9

add 81 to both sides:

x² + 18x + 81 = -9 + 81

Factor the left side

(x + 9)(x + 9) = 72

Write the left side as the square of a binomial:

(x + 9)² = 72

Use the principle of square roots.  

1. Eliminate the square and the parentheses on the left.
2. Take ± the square roots of the right side:

  x + 9 = 

  x + 9 = 

  x + 9 = 

      x = -9 ± 6

Two solutions:

       x = -9 + 6 and       x = -9 - 6

 Edwin