Completing the Square and the Quadratic Formula

(1) We now have three of the five methods for solving quadratic equations – can you name and demonstrate each of them? We’ll now take on the task of learning the last two methods .

(2) The first of these relies on observing how simple it was to solve quadratic equations of the form (x-4)²=7 by using the Perfect Square Rule. The left side of this equation is the square of a binomial difference and would look like x²-8x+16 if expanded. We would recognize this as a perfect square trinomial based on the hundreds of factoring problems we’ve done. So, had the equation been given as x²-8x+16 =7 , we could factor the left side and get (x-4)²=7 and then use the Perfect Square Rule. But what if we were given x²-8x+9=0 ? The observant student should see how I got this – subtract 7 from both sides of x²-8x+16 =7  . Our quest is to do the reverse of this – start with x²-8x+9=0 and discover how to manipulate it into a perfect square trinomial equaling a number.

(3) The method is called completing the square, and it relies on understanding the form of a perfect square trinomial a bit better:
(x+a)²=x²+2ax+a²
Notice the linear coefficient, 2a. Taking one half of it, , gives the object which is added to x in the binomial , x+a, and whose square, a², appears as the constant in the trinomial. So that is how perfect square trinomials are formed: the constant is the square of1/2 of the linear coefficient .

(4) We use the above concept to create a perfect square trinomial by the technique called completing the square.

Example 1: Solve x²+6x-2 = 0
Solution: In order to create the proper constant to have this be a perfect square trinomial , we first make room for it by, in this case, adding 2 to both sides.
 

	Make room for new constant
	Take 1/2 the linear coefficient The square of is the constant we need to make a perfect square trinomial, x2 + 6x + 9 But we can't just add to one side, we have to add it to BOTH side
	Simplify
	Use Perfect Square Rule
	Subtract 3 from both sides

 Example 2: Solve x²-5x+3=0
Solution:

	Make room for new constant
	Take half the linear coefficient
	The square of this is the new constant which we add to BOTH sides
	Use Perfect Square Rule
	Add 5/2 to both sides

(5) ZZZZZ This method ONLY WORKS ON MONIC QUADRATICS – i.e., where the leading coefficient is 1. If it isn’t 1 in what you start with, then you must divide both sides of the equation by the leading coefficient to make monic.

Example 3: Solve 4x²-8x-3=0
Solution:

	Make monic by dividing by 4
	Create constant space
	, whose square is added to BOTH sides
	Use Perfect Square Rule
	Add 1 to both sides

(6) We can apply this method to an arbitrary quadratic equation in standard form. The end result is a formula for x in terms of the coefficients a, b and c. It is known as the Quadratic Formula .

Quadratic Formula

are the solutions to the equation
ax²+bx+c=0.where a≠0   

(7) Let’s concentrate a bit on the a, b and c in the standard form of the quadratic equation.

This statement highlights something more than just the definitions of a, b and c. It says there can be ONLY ONE term containing x², there can be ONLY ONE term containing x, and the constant is the entire collection of terms which do not contain x. For example, consider mx²+4x=3-mx. This is quadratic in x so we create an equation to 0: mx²+4x+mx-3=0. This is NOT in standard form since there are two linear terms, 4x and mx. We can ONLY have ONE linear term in order to use the Quadratic Formula, so we create a single linear term by distributing an x out of these two terms: mx²+(4+m)x-3=0 . Now we can identify a, b and c: a=m,b=4+m,and c=-3. Thus, we have


Here’s another example: Solve for x in (x-1)²+y²=4x. Notice first that we can’t isolate the (x-1)² as 4x-y² and use the Perfect Square Rule since this would place an x in the answer to a problem where we want to solve for . So we go back to basics and multiply it out:

	Remove grouping symbols
	Quadratic in x=> try for standard form
	OK:a=1,b= -6,c=1+y2
	That's right, remember is made up of all terms not containing x