Answer:
To complete the square for the equation x² - 13x + 31 = 0, we need to add a number to make the expression (x - a)², where a is half of the coefficient of x.
The coefficient of x in the equation is -13. Half of -13 is -13/2 = -6.5.
So, we want to rewrite the equation in the form (x - (-6.5))². Squaring -6.5 gives us 42.25.
However, our original equation has a constant term of 31. So, we need to subtract 31 from 42.25 to find out what number needs to be added to the equation to complete the square.
42.25 - 31 = 11.25
So, the number that needs to be added to the equation x² - 13x + 31 = 0 to complete the square is 11.25.