Question

The quadratic x^2+440x+440^2 can be written in the form (x+b)^2+c, where b and c are constants. What is c/b?

Answer 1

This form (x+b)²+c is a vertex form that can be found by completing square for the equation x²+440x+440².

To complete square we will need to use formula a² +2ab + b² = (a+b)²

x²+440x+440²,
x² +2*220²x +220² -220² +440² (we added 220² -220², so nothing really changed here).

x² +2*220²x +220² -220² +440² ( here , bold part can be rewritten as (x+220)²)

(x+220)² -220² +440² (calculating bold part)

(x+220)² -220² +440² = (x+220)² + 145200

Now we have x²+440x+440², in the form that we need
x²+440x+440² = (x+220)² + 145200

(x+b)²+c = (x+220)² + 145200,
b=220,
c= 145200

So,
c/b = 145200/220 = 660
c/b = 660/1 (if you need it in a fraction form) or 660.