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.