Why 'Completing the Square' is the Best Way to Solve Polynomials
(This is a blog post aimed at Higher GCSE and A-level content in the UK.)
I am a man on a mission.
The mission is simple...
To convince the world that completing the square is objectively the best way to deal with any quadratic polynomial.
What is a polynomial?
If you aren't familiar, a polynomial refers to most equations you might come across in maths, like those below:
(The 'degree' refers to the biggest power of x in each equation. Notice that a 'degree 5' equation starts with the term '-2x⁵' — it's that ⁵ bit that means it's a 'quintic' equation. The 'degree' of the equation, from linear to quadratic to cubic and beyond, determine the overall shape of the graph.)
In GCSE maths, however, we usually only bother with linear equations (e.g., y = 2x + 5) or quadratic equations (e.g., y = x² + 5x + 4).
Did you know? Quadratic equations are called 'quadratic' because of the Latin term 'quadratus', meaning 'square'. Quadratic equations are any equations in which the greatest power of x is x², pronounced 'x-squared'. Neat, huh?
How can I solve polynomials?
For quadratic polynomials, you have three methods of 'solving' them — that is, for finding the value of x that makes
**EQUATION** = 0
Those three methods are:
- Factorising — turning an equation like x² + 5x + 4 into the product of two brackets, like (x + 1)(x + 4). Factorising is the most commonly encouraged process, and I won't go through how to do it here.
- Completing the Square — turning an equation like x² + 5x + 4 into (x + 5/2)² — 9/4. This is a more niche process that teachers seem to gloss over in class.
- The Quadratic Formula. Yes, that equation — check it out!
For solving a quadratic, each of these methods works:
When factorising into the form (ax + b)(cx + d), you can read off the roots as being at x = -b/a and -d/c. This is because the two ways for the product of two brackets to be zero are simply for either bracket to equal 0. In mathematical terms,
Option 1 for (ax+b)(cx+d)=0: ax+b = 0, so ax = -b, so x = -b/a
Option 2 for (ax+b)(cx+d)=0: cx+d = 0, so cx = -d, so x = -d/c
But what do you need to do before you even get to this step? You need to trial and error numbers that multiply to make the last number of the quadratic and add up to make the middle number! And it's not always obvious, especially when those numbers aren't integers.
So, yes, quadratics give you your solutions (a.k.a. roots) really easily, but getting to that point in the first place is often the hard bit - especially with harder questions using fractions, decimals or larger numbers!
So what can you do instead?
Well, the Quadratic Formula has a nasty reputation, but it's really quite easy. All you do is... plug loads of numbers into an equation and either solve it slowly by hand, or stick it painstakingly into a calculator and hope you don't mis-type anything.
If you're computing it by hand, the number of actual number of steps in doing the Quadratic Formula is generally greater than the number of steps required for either of the other two methods. Meaning it's all round a less-than-efficient approach to solving your quadratic.
Plus, the other downside is that the Formula finds you just the two roots. It's inflexible, and, to my mind, un-fun. Plugging numbers into a calculator is not the mathematician's way!
That's where Completing the Square comes in.
Once you have an equation in the form a(x - b)² - c, you can do anything you like with it.
The turning point of the equation can be read off at x = b (rendering the entire bracketed term 0) and y = -c. That means the minimum (or, for negative quadratics, maximum) point of the graph is at (b, -c).
The reason for this is that a squared term, like 3² or (-3)² or even (x - b)² must always be greater than or equal to zero. You cannot square a (real) number and get anything less than zero! Try it.
The roots can also be solved by a simple three-step rearrangement of a(x - b)² - c = 0
Step 1: (x - b)² = c/a
Step 2: x - b = ±√c/a
Step 3: x = b ± √c/a
This means that the completed-square form allows you to figure out the turning point and the roots of the equation with very minimal effort.
Editor's note: You can, of course, find the turning point from any of the other methods too, since parabolas (the shapes made by quadratic equations when graphed) are always symmetrical. That means that the roots where the graph crosses the x-axis will be evenly spaced around the turning point. So, if you find the midpoint of the x-values that make y = 0, you'll have found the x-value of the turning point too!
Why is Completing the Square so good?
The real benefit, however, to completing the square, is that you can complete the square on anything. Literally, any quadratic you come across will be 'completable' using this method.
On the other hand, very many quadratics cannot be factored. Take x² + x + 10, for instance. There are no real numbers that will multiply to make 10 and add to make 1 (for the single, lonesome x in the middle of the equation).
If you put those coefficients (a = b = 1 and c = 10) into the Quadratic Formula, you'll end up square-rooting a negative number (try it!) and your calculator will hate you.
On the other hand, you can still complete the square, find the turning point of the graph, and sketch it on some axes — and, by doing so, you'll notice why factorising and the Quadratic Formula didn't work! It's because the equation has no roots at all!
The universality of completing the square cannot be beaten.
Factorising may occasionally be convenient in easier styles of question, but this is, in my opinion, misleading. The easiness of factorising makes students want to factorise everything - to the point that they may forget that C.T.S. exists!
For Higher GCSE, A-level, and beyond, C.T.S. reigns - all the way.
Challenge: By considering the quadratic equation ax² + bx + c, can you complete the square on this algebraically? (If you do so, you'll notice another reason why the Quadratic Formula is subservient to C.T.S. — because that's how the Quadratic Formula was derived in the first place!)
Exam Tip: When Completing the Square, always use fractions, never decimals. It's so much easier to square a fraction like 5/2 into 25/4 (= 5²/2²) than it is to square the equivalent decimal 2.5 into 6.25.
Thought of the Day: Why is 'Completing the Square' called 'Completing the Square'? In the example (x - 3)² + 5, what is the 'square' bit, and which bit is 'completing' it, do you think?