Completing the square is a process by which we turn an expression of the form ax ² + bx + c into an expression of the form a(x + h)² + d. The usefullness of this technique was shown in lecture on Tuesday evening in finding the center and radius of a circle from an equation for a circle that was not in standard form.
First if a is not one, then we factor out a , so we have a[x ² + (b/a)x] + c . To make notation easier, let r = b/a . Then we want to write a[x ² + rx] + c in the form a(x + h)² + d .
The completing the square algorithm tells us that we let h = r/2 and we write a[x ² + rx] + c = a[x ² + rx + h² - h ² ] + c = a[(x + h) ² - h² ] + c = a(x + h) ² + (c - a h² ).
How this relates to the quadratic formula
Assume that we want to write x ² + bx + r as (x + h) ² + d . In order to do this, we need to find a numbers c, h such that x ² + bx + c = (x + h) ² . We can find these numbers by realizing that the two expressions must have the same roots. The expression (x+h) ² has only one root, -h , with multiplicity two. To find the roots of x ² + bx + c , we can use the quadratic formula. Since we would like on root with multiplicity two, we know that the part of the quadratic formula under the square root, b ² - 4ac (the discriminant) has to be zero. In this case a = 1 , so we need b ² = 4c and c = b ² /4 = [b/2] ² . Then, from the quadratic formula, h = -b/(2a) = -b/2
Therefore, x ² + bx + (b/2) ² = (x - (b/2)) ² . Finally, x ² + bx + r = (x - (b/2)) ² - (b/2)² + r . If you compare this with the result described above, you will not that it is the same. I would not suggest trying to remember all this, but it is always interesting to look into the relationships between math concepts.