what number must be added to the following expression in order to complete the square?
I don't see an equation. However, if your equation is in the form: 1x^{2} + bx = 0, in order to make sure the equation is a perfect square, divide b by 2, square it, and add to both sides.
Example: x^{2} + 6x = 0, b = 6, b/2 = 6/2 = 3, so (b/2)^{2} = 3^{2} = 9.
So add 9 to both sides => x^{2} + 6x + 9 = 9, and (x + 3)^{2} = 9.
Another example: 3x^{2} - 15x - 12 = 0; divide by 3: x^{2} - 5x - 4 = 0; add 4: x^{2} - 5x = 4;
Take 1/2 of -5, square it, and add to both sides: x^{2} - 5x + (25/4) = 4 + (25/4)
This becomes: (x - (5/2))^{2} = (16/4) + (25/4) => (x-(5/2))^{2} = 41/4.
I hope this helps.