Solving x^2+bx+c=0 by Factorising

Picture a ball thrown straight up out of your hand. It climbs, slows, turns over, and falls back down past you. Now ask a simple question about it: when was that ball exactly five metres off the ground? There are two moments, not one. Once on the way up, and once on the way back down, and both of them are honest answers to the question. The maths that describes a flight like that is a quadratic, and a quadratic hands you two answers on purpose. Stop at the first one and you have only described half the journey.

So before any method at all, one rule about these equations, because everything else today hangs off it. An equation with an x squared in it normally has two solutions rather than one. Both are genuinely correct, and a question that says solve is asking you for both of them. Here is your handle for this video: two brackets, two answers. That is the line to carry into the exam hall, and it will be doing real work in a minute. Now, why two? The reason is one small fact about zero, and that fact does all of the heavy lifting here. If you multiply two numbers together and the result is zero, then one of them has to have been zero. Three times seven is twenty-one. Three times four is twelve. Nothing you multiply together lands on zero unless something in it was zero already. So when a quadratic sits there as two brackets multiplied together, equalling zero, either the first bracket is zero or the second bracket is zero. That is two separate ways of making the equation true, and that is exactly where your two answers come from. Quick check before we go further. Take bracket x plus five, times bracket x minus two, equals zero. Which value of x makes that first bracket, x plus five, come out as zero: is it five, or is it minus five? Pick one. I'll wait. The answer is minus five. Minus five plus five is nothing at all, so that first bracket collapses to zero, and once one bracket is zero the whole product is zero whatever the other bracket happens to be. One bracket hitting zero is enough to make the whole thing zero, and that is the engine inside every example coming up.

Time to run that engine on a real equation, from the first line to the last. Solve x squared plus three x minus ten equals zero. It already sits in the shape we want, with a plain x squared at the front and a zero on the right-hand side. That tidy shape matters. When an equation is not in it yet, the video on rearranging and factorising a general quadratic equation is the one that gets it there. Factorising the left-hand side is the skill this builds on, so here it is quickly rather than at length. You want two numbers that multiply to make minus ten and add to make three. Plus five and minus two do both jobs at once. So the equation becomes bracket x plus five, times bracket x minus two, equals zero. Same equation, written as a product. Here is the step where second answers go missing. Do not set the whole factorised thing to zero in one lump. Take each bracket on its own line, and give each one its own small equation. First line: x plus five equals zero, which gives x equals minus five. Second line, written underneath it: x minus two equals zero, which gives x equals two. Two brackets, two answers. x equals minus five, or x equals two. Write both of them down with the word or sitting between them, because either value solves the equation on its own. It is worth thirty seconds to watch both of them work. Put minus five in: twenty-five, take away fifteen, take away ten, comes to zero. Put two in: four, add six, take away ten, comes to zero as well. Neither one is the real answer with the other as a spare. The equation is genuinely true at both values, and a solution you left out is a solution you got wrong.

The next equation is the one that catches people out, because something is missing from it. Solve x squared minus sixteen equals zero. There is no x term in the middle at all, and that missing middle is exactly where the two-answer habit tends to slip. It still factorises into two brackets. You are after two numbers that multiply to make minus sixteen, and that add to make nothing. Pause it there and work it out. I'll wait. The answer is x equals four, or x equals minus four. The brackets are x minus four and x plus four, and minus four x and plus four x cancel each other out, which is why there was no middle term in the first place. Four is the one people write down; minus four is the one people forget, and minus four squared is sixteen just as surely as four squared is. Whenever the middle term is missing like that, your two answers are the same number with opposite signs, and both of them belong on the page.

Now the exam side of this, and one working habit that quietly costs students the second answer. Plenty of students hunt for a solution by trying numbers until one of them works. The moment a number works, they stop, because as far as trial and improvement is concerned the job is finished. Here is a line from an examiner report, on a question where a quadratic had to be solved. Pupils often used trial and improvement and did find the answer of 4; but failed to recognise that a quadratic should have two solutions. Those that attempted to factorise often made errors but many did know to equate the brackets to zero. Listen to that last sentence again. Even the students whose factorising went wrong still knew to set each bracket to zero, and that single habit is what protects the second answer. That is the real reason to factorise rather than hunt for numbers. Trying values can stop the instant one of them works. Factorising cannot stop early, because it hands you both brackets at the same moment, so both answers are already sitting in front of you. So if your working ends with a single value of x, treat that as a signal to go back and check whether a bracket went unused.

One last one for you, and this time there is nothing on screen to copy from. Solve x squared minus x minus six equals zero. Same shape as before, so factorise the left-hand side, then set each bracket to zero on a line of its own. Take your time. I'll wait right here. The answer is x equals three, or x equals minus two. The brackets are x minus three and x plus two, because minus three and plus two multiply to give minus six and add to give minus one. If you found one of those and stopped there, that is the precise habit this video exists to break.

Right, the whole method in three steps, and then you are done. Step one: factorise the left-hand side into two brackets multiplied together. Step two: set each bracket equal to zero, each on its own line. Step three: write both values of x, with the word or between them. Two brackets, two answers. And when the middle term is missing, those two answers are the same number with a plus and a minus in front of it. A quadratic equation is asking you for two solutions, so an answer with one value of x in it is an answer that is still half finished.

Next in the chapter: Rearranging and Factorising a General Quadratic Equation. That one takes the equations that are not tidy yet and gets them into the shape you have just been solving.

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