completing the square and the quadratic formula

Question

Discuss the relationship between completing the square and the quadratic formula

Answer 1

If you take the standard form of the equation

 

ax² + bx + c = 0

 

and you complete the square, you derive the quadratic formula. 

 

Move the C and factor out the a to

 

a(x²+[b/a]x )=-c

divide both sides by a and take half the middle term and square it and add to both sides to come up

x2+[b/a]x+[b²/(4a²)]=-c/a+b²/4a²

Then it becomes a perfect square on the left side and common denominator on the right is simplified 

(x+(b/2a))²=(b²-4ac)/4a²

taking the square root of both sides

x+b/2a=+/- [sqrt(b²-4ac)]/2a

Then subtract the b/2a and we then have the quadratic formula.