Rearranging and Factorising a General Quadratic Equation

Picture a rectangular lawn you have been asked to turf. It covers thirty-two square metres, and it is four metres longer than it is wide. Nobody has told you the width. You can turn that into an equation in about ten seconds. Solving it is where you stop, because the equation that comes out has no zero anywhere near it. Every method for cracking a quadratic expects that zero to be sitting there already, and a real problem almost never hands it to you that way. So there is a step before any solving: dragging the equation into the shape the method needs. That step is what this video is about.

A quick word on who this is for. It is Higher tier material. Foundation students want the video on solving a quadratic that already equals zero by factorising instead.

Start with a plain equation, with no story attached to it at all, so that the rearranging itself is the only thing on the screen you need to watch.

Here it is. x squared plus seven x equals eighteen. That is a quadratic, and it is definitely solvable, but not yet. The reason a quadratic has to equal zero comes down to one fact about zero. If two numbers multiply to give zero, at least one of them has to be zero. No other number behaves like that. Eighteen has no such power. Two things multiplying to eighteen could be nine and two, or six and three. Knowing the product tells you nothing at all about either bracket. So factorising the left side as it stands, into x times x plus seven equals eighteen, gets you nowhere. Neither bracket has to equal anything in particular. The zero is not a formality there; it is the entire reason the method works at all.

So everything goes onto one side. Subtract eighteen from the left, and subtract eighteen from the right, so the balance never breaks. On the right, eighteen minus eighteen leaves nothing but zero. On the left, the eighteen lands as minus eighteen. x squared plus seven x minus eighteen equals zero.

Say this to yourself while you do it: cross the line, flip the sign. Any term that moves across the equals sign arrives on the other side with its sign flipped.

Plus seven x stays exactly as it is, because it never moved. Only the eighteen crossed, so only the eighteen changed. Sign slips creep in when people flip everything out of habit.

Now it is a shape you already know how to handle. Two numbers that multiply to give minus eighteen and add to give plus seven: that is plus nine and minus two. So it factorises as x plus nine, times x minus two, equals zero. One of those brackets has to be zero, so x is minus nine, or x is two.

Your turn, and it is only the rearranging. Take x squared equals five x plus six. Three people rearrange it into standard form. A gets x squared plus five x plus six equals zero. B gets x squared minus five x minus six equals zero. C gets x squared minus five x plus six equals zero. Only one of them has carried both signs across correctly. Which one is it? Take your pick. I'll wait. It is B: x squared minus five x minus six equals zero. Both terms crossed the equals sign, so both of them flipped. Five x became minus five x, and plus six became minus six. A flipped nothing at all. C flipped the five x but left the six alone, and that is the easy one to miss, because a lone number looks harmless. And it is in the right shape now, so finish it off: minus six and plus one multiply to minus six and add to minus five. x is six, or x is minus one.

Back to that lawn now, because most of the time nobody hands you an equation at all. You have to build it yourself.

Call the width x metres. The length is four metres longer, so the length is x plus four. Area is length times width, so x times x plus four equals thirty-two. There is no zero, and there is a bracket in the way. Expand the bracket first, and you get x squared plus four x equals thirty-two. Same move as before. Sweep the thirty-two across, tidy it into standard form, and factorise it before I do. Pause it there and work it out. I'll wait. Thirty-two crosses the line, so it changes sign. x squared plus four x minus thirty-two equals zero. Two numbers multiplying to minus thirty-two and adding to plus four are plus eight and minus four. So x plus eight, times x minus four, equals zero, which gives x equals minus eight, or x equals four. A lawn cannot be minus eight metres wide, so that answer gets thrown away. The width is four metres. Worth five seconds to check it: four metres wide, eight metres long, thirty-two square metres of turf. It fits.

Now the case where factorising simply refuses to work, and what you reach for instead.

Take x squared minus six x plus four equals zero. This one already ends in equals zero, so there is no rearranging to do. The trouble starts one step later. You need two whole numbers that multiply to four and add to minus six. Only four pairs multiply to four, and their sums come out as five, four, minus five, and minus four. There is no pair. Not because you have missed one, but because the answers to this equation are not whole numbers in the first place. That is your signal to complete the square. It rewrites the left side as one bracket squared, plus or minus a number, and that matters because a squared bracket can be square-rooted. Three separate terms cannot. Halve the number in front of x. Half of minus six is minus three, so the bracket you want is x minus three, all squared. But x minus three, all squared, multiplies out to x squared minus six x plus nine. Nine, when we only want four, so it is five too many. Subtract that five back off, and the left side becomes x minus three, all squared, minus five. So the equation becomes x minus three, all squared, minus five, equals zero. Add five to both sides, and x minus three, all squared, equals five. Square-root both sides, and here is where the second answer comes from: a square root can be positive or negative. So x minus three is plus or minus the square root of five. Add three to both sides and you are finished. x equals three plus the square root of five, or three minus the square root of five.

Now, how to write that down. If the question says give your answer in surd form, or leave your answer exact, then three plus or minus root five is the finished answer. The calculator stays shut. If it asks for two decimal places instead, then you round, and only then: five point two four, and nought point seven six. The question decides the form, not you.

Time to look at how these arrive on the paper, because there is an instruction sitting in the question that people read straight past. Look at the wording itself. Solve by factorising. Solve by completing the square. Give your answer in surd form. Those words are not decoration; they name the method you are required to use. Here is one line from an examiner's report, about a question where candidates were told to factorise. Many candidates did not factorise the equation as directed and used alternative methods such as the quadratic formula, completing the square or trial and improvement. Directed is the word doing the work there. The mark scheme is built around the named method, so an answer that looks right, reached another way, can still fall short. There is a bonus buried in that instruction, though. If a question tells you to factorise, it is promising you that the equation does factorise. So when you cannot find the pair of numbers, go back and check your rearranging, because that is usually where a sign went astray. Examiners describe that same slip in another report. This one covers a question that needed a different method, so listen for the last sentence. Many students did not know that the solutions needed the application of the quadratic formula. Those who did, often could not remember the correct formula. Sign errors were made when rearranging the equation and there were substitution errors when the correct formula was used. Sign errors were made when rearranging the equation. That sentence is ours, and it is why we spent so long moving one number across one equals sign. The formula those students were reaching for has a video of its own, so it stays out of this one.

Cross the line, flip the sign. That is the one to keep, and here is the whole method built around it, start to finish. One. Get every term onto one side, until the other side is nothing but zero. If your equation does not end in equals zero, you have not finished rearranging. Two. Every term that crosses the equals sign flips its sign, and the terms that never moved do not change at all. Three. Factorise if it factorises. If no pair of whole numbers works, complete the square instead, and expect an answer with a root in it. Four. If the equation came from a real situation, hold both answers up against it, and throw away the one that cannot happen, like a negative width. And if the question names a method, that is the method you use, however tempting anything else looks.

Next in the chapter: Solving a Quadratic with the Quadratic Formula.

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