How is quadratic formula different than completing the square?
1 Answer
They are closely related, but you might choose to use one over the other for a specific purpose, e.g. finding the vertex of a parabola rather than its
Explanation:
Given
a(x+b/(2a))^2 + (c - b^2/(4a)) = 0a(x+b2a)2+(c−b24a)=0
Then subtract
a(x+b/(2a))^2 = b^2/(4a) - c = (b^2-4ac)/(4a)a(x+b2a)2=b24a−c=b2−4ac4a
Divide both ends by
(x+b/(2a))^2 = (b^2-4ac)/(4a^2) = (b^2-4ac)/(2a)^2(x+b2a)2=b2−4ac4a2=b2−4ac(2a)2
Taking square roots we get:
x+b/(2a) = +-sqrt(b^2-4ac)/(2a)x+b2a=±√b2−4ac2a
Finally subtract
x = -b/(2a)+-sqrt(b^2-4ac)/(2a) = (-b+-sqrt(b^2-4ac))/(2a)x=−b2a±√b2−4ac2a=−b±√b2−4ac2a
If we just want to get straight to the roots then the quadratic formula is probably the best choice.
If we want to get to a vertex form to find where the vertex of the parabola is, then completing the square is more appropriate.
Of course, one advantage of completing the square over the quadratic formula is less 'magic'. It is much clearer how you are getting to the solution and it does not entail much more work. However, now you can derive the quadratic formula yourself you have every reason to use it if it's more convenient.
