Completing the Square
Say we have a simple expression likeÂ x^{2}Â + bx. HavingÂ xÂ twice in the same expression can make life hard. What can we do?
Well, with a little inspiration from Geometry we can convert it, like this:
As you can seeÂ x^{2}Â + bxÂ can be rearrangedÂ nearlyÂ into a square …
… and we canÂ complete the squareÂ withÂ (b/2)^{2}
In Algebra it looks like this:
x^{2}Â + bx  + (b/2)^{2}  =  (x+b/2)^{2} 
“Complete the Square” 
So, by addingÂ (b/2)^{2}Â we can complete the square.
AndÂ (x+b/2)^{2}Â hasÂ xÂ onlyÂ once, which is easier to use.
Keeping the Balance
Now … we can’t justÂ addÂ (b/2)^{2}Â without alsoÂ subtractingÂ it too! Otherwise the whole value changes.
So let’s see how to do it properly with an example:
Start with:  
(“b” is 6 in this case)  
Complete the Square:  

AlsoÂ subtractÂ the new term 
Simplify it and we are done. 

The result:
x^{2}Â + 6x + 7 Â = Â (x+3)^{2}Â âˆ’Â 2
And nowÂ xÂ only appears once, and our job is done!
A Shortcut Approach
Here is a quick way to get an answer. You may like this method.
First think about the result we want:Â (x+d)^{2}Â + e
AfterÂ expandingÂ (x+d)^{2}Â we get:Â x^{2}Â + 2dx + d^{2}Â + e
Now see if we can turn our example into that form to discover d and e
Example: try to fitÂ x^{2}Â + 6x + 7Â intoÂ x^{2}Â + 2dx + d^{2}Â + e
Now we can “force” an answer:
 We know thatÂ 6xÂ must end up asÂ 2dx, soÂ dÂ must be 3
 Next we see thatÂ 7Â must become d^{2}Â + e =Â 9 + e, soÂ eÂ must be âˆ’2
And we get the same resultÂ (x+3)^{2}Â âˆ’ 2Â as above!
Now, let us look at a useful application: solving Quadratic Equations …
Solving General Quadratic Equations by Completing the Square
We can complete the square toÂ solveÂ aÂ Quadratic EquationÂ (find where it is equal to zero).
But a general Quadratic Equation can have aÂ coefficientÂ ofÂ aÂ in front ofÂ x^{2}:
ax^{2}Â + bx + c = 0
But that is easy to deal with … just divide the whole equation by “a” first, then carry on:
x^{2}Â + (b/a)x + c/a = 0
Steps
Now we canÂ solveÂ a Quadratic Equation in 5 steps:
 Step 1Â Divide all terms byÂ aÂ (the coefficient ofÂ x^{2}).
 Step 2Â Move the number term (c/a) to the right side of the equation.
 Step 3Â Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
We now have something that looks like (x + p)^{2}Â = q, which can be solved rather easily:
 Step 4Â Take the square root on both sides of the equation.
 Step 5Â Subtract the number that remains on the left side of the equation to findÂ x.
Examples
OK, some examples will help!
Example 1: Solve x^{2}Â + 4x + 1 = 0
Step 1Â can be skipped in this example since the coefficient of x^{2}Â is 1
Step 2Â Move the number term to the right side of the equation:
Step 3Â Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.
(b/2)^{2}Â = (4/2)^{2}Â = 2^{2}Â = 4
Step 4Â Take the square root on both sides of the equation:
Step 5Â Subtract 2 from both sides:
And here is an interesting and useful thing.
At the end of step 3 we had the equation:
It gives us theÂ vertexÂ (turning point) of x^{2}Â + 4x + 1:Â (2, 3)
Example 2: Solve 5x^{2}Â â€“ 4x â€“ 2 = 0
Step 1Â Divide all terms by 5
Step 2Â Move the number term to the right side of the equation:
Step 3Â Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:
(b/2)^{2}Â = (0.8/2)^{2}Â = 0.4^{2}Â = 0.16
Step 4Â Take the square root on both sides of the equation:
Step 5Â Subtract (0.4) from both sides (in other words, add 0.4):
Why “Complete the Square”?
Why complete the square when we can just use theÂ Quadratic FormulaÂ to solve a Quadratic Equation?
Well, one reason is given above, where the new form not only shows us the vertex, but makes it easier to solve.
There are also times when the formÂ ax^{2}Â + bx + cÂ may be part of aÂ largerÂ question and rearranging it asÂ a(x+d)^{2}Â +Â eÂ makes the solution easier, becauseÂ xÂ only appears once.
For example “x” may itself be a function (likeÂ cos(z)) and rearranging it may open up a path to a better solution.
Also Completing the Square is the first step in theÂ Derivation of the Quadratic Formula
Just think of it as another tool in your mathematics toolbox.