Solving Quadratic equations by Completing the Square

quadratic formula, completing the square, perfect square, second degree polynomial, square root

The quadratic formula which is well known by all students taking advanced Algebra is actually rooted in the more fundamental ideas of the completing the square.  Thus, Completing the square to slove quadratic equations is actually more important than any other method.

Quadratic Equations are second degree polynomials with the general form as follows:

ax²+bx+c=0

We complete the square on this form as follows.

ax²+bx+c=0

ax²+bx=-c [Subtracting "c" from both sides]

(ax²)/a+(bx)/a=(-c/a) [Dividing by "a" on both sides]

x²+(b/a)x=(-c/a)  [rewriting the line above]

x²+(b/a)x+(b/2a)²=(-c/a)+(b/2a)² [adding the square of half the coefficient of x to both sides]

now on the left, we have a competed perfect square, so:

x²+(b/a)x+(b²)/4a²=(-c/a)+(b²/4a²)

(x+(b/2a))²=(-c/a)+(b²/4a²)

(x+(b/2a))²=(-4ac/4a²)+(b²/4a²)

(x+(b/2a))²=(b²-4ac)/(4a²)

(x+(b/2a))=±((b²-4ac)/(4a²)) [taking the square root of both sides]


(x+(b/2a))=±((b²-4ac))/(2a)

(x)=(-b/2a)±((b²-4ac))/(2a)


(x)=(-b±((b²-4ac))/(2a) [quadratic formula]

Here, we get the competed square which leaves us with the quadratic formula, when we complete the square, it merely requires to follow this outline and it essentially amounts to using the quadratic formula.

quadratic formula, completing the square, perfect square, second degree polynomial, square root