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Completing The Square

Updated: Jun 16, 2021

Dear Secondary Math students, Math Lobby will cover Completing The Square which is widely used in both Additional Math and Elementary Math for both the O Level and N level Math examination. Without further ado, let's begin!



In this note, you will learn:


1) Derivation of completing the square and how to solve a quadratic equation


2) What can this technique help us find


1) Derivation of completing the square


Given an example of the quadratic equation:


y = x² + bx


To solve for an equation with a is more difficult and tedious as compared to just having a single x in the expression. Hence, we can utilize the technique of “completing the square” to help us simplify the equation.




From this diagram, you can see that in order to complete the square, we added a

(b/2)².


Therefore, the initial equation after “completing the square” will look like this:


y = x² + bx + (b/2)² = (x + (b/2))²



In this form, there is only x present in the equation, which is easier to work around with.

However, this constant was not part of our equation to begin with. Hence, we must subtract it off the equation in order to make the equation hold true.



Therefore, the final equation will be:


y = (x + b/2)² - (b/2)² = (x + b/2)² - b²/4

which can then be re-written as:


y = (x - d)² + e, where d = b/2a , e = c - b²/4a



Try it now with Math Lobby!

Question: Express y= x²+6x in complete the square form. Let us know your answer in the comments section below!


Now, you might be wondering: “Shouldn’t the variable, d & e, be equal to and - respectively?”


The reason is because for simplicity sake, we used the equation y = x² + bx, but many a times, there will be a constant in front of the x² variable and in the equation. Hence, a typical quadratic equation will look like this:



ax² + bx + c



So, how do we “complete the square” and solve the equation in this manner? It is actually very simple, just follow the steps below:



1. Divide all the terms by a (the coefficient of ), which will give us:


x² + (b/a)x + c/a = 0



2. Shift the number term, c/a , to the right side of the equation.


x² + (b/a)x = -c/a


Now, do you guys spot something familiar on the left side of the equation?



Yes! That is how we learnt to complete the square back in the first part of the introduction, so now we can apply that same concept into this equation.



3. Complete the square on the left side of the equation and balance this by adding the same value on the right side of the equation.


x² + (b/a)x = -c/a


(x + (b/2a))² = -c/a + (b/2a)²


Multiply the whole equation by a:


a(x + b/2a )² = -c + b²/4a


4. Take the square root on both side of the equation



5. Solve for x!



That’s it! You’ve just learnt how to solve for x using the “completing the square” technique!



Try it now with Math Lobby!

Question: Solve 2x²+8x-10 using completing the square method. Let us know your answer in the comments section below!


Math Lobby Bonus Tip: What can this technique help us find


In graphs and functions, this “completing the square” technique can be very useful in finding a number of things:


· x and y coordinates of a maximum turning point


· x and y coordinates of a minimum turning point



Depending on the coefficient of the variable,


If it is positive: the quadratic function’s curve will have a minimum turning point


If it is negative: the quadratic function’s curve will have a maximum turning point


To find the maxima and minima, we will have to recognize a way of expressing the quadratic equation:



a(x + b/2a+ c - (b²/4a)


y = a(x - h)² + k, where h = -b/2a and k = c - b²/4a



Hence, once we have “completed the square” of a quadratic equation, we can simply know the vertex of the turning point.


h = x-coordinates of the turning point


k = y-coordinates of the turning point


And that is how you find the vertex of a quadratic curve!



That’s all for today, and I hope that everyone has a clear basic concept of the derivation of how “completing the square” came about, a good grasp of how to solve a quadratic equation using “complete the square” technique, and how we actually use the “complete the square” form to find the vertex of a quadratic equation.



Try it now with Math Lobby!

Question: Find the turning point of 4(x-5)²+3. Determine whether is it a minimum or maximum point. Let us know your answer in the comments section below!


Work hard, stay motivated and we wish all students a successful and enjoyable journey with Math Lobby!



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