The formula this book hands you for the roots, and how to apply it

0:00Up to now in this subject, everything has been earned. You were shown why a factorisation works, and why a root is a zero. And then a formula arrives. x equals minus b, plus or minus the square root of b squared minus four a c, all over two a. It is correct. It works. And almost nobody is shown where it came from. So this video will read it slowly, and then derive it, in four lines.

0:31The derivation is not decoration. It hands you a condition the bare formula leaves out, and it tells you where the plus-or-minus comes from. Start with the reading, because there is a real trap in it. The whole of minus b plus or minus the root sits over two a — one fraction bar, spanning everything above it. The commonest way to write it down badly is to let the two a divide only the square-root part, leaving the minus b outside.

1:06That is not a small slip. On x squared plus seven x minus sixty equals nought, the honest formula returns five and minus twelve; the badly written one returns three halves and minus thirty-one halves. Neither of those is a root. Put three halves in and you get two hundred and eighty-seven quarters, which is nowhere near nothing. The thing under the root sign is called the discriminant, and it is b squared minus four a c. Everything interesting here happens to that quantity.

1:45Now the condition, which is the part most often dropped. The formula is only good for real roots while the discriminant stays at nought or above — because the square root of a negative number is not a real number, so there is nothing for the formula to return. That condition is also easy to confirm without testing anything. Decide whether a real root EXISTS by looking at the discriminant, and you have proved nothing.

2:16So it was tested the other way round. Five thousand quadratics were swept, each worked out at hundreds of points and asked whether it ever reached nought or crossed it. That search never forms b squared minus four a c. The two verdicts agreed on all five thousand. And to be sure that meant something, the table was built again with the sign flipped — b squared PLUS four a c. That one disagreed two thousand seven hundred and seventy-six times.

2:51Now the working, and it starts with a move that looks like it comes out of nowhere. Take a x squared plus b x plus c equals nought, and multiply everything by four a. That gives four a squared x squared, plus four a b x, plus four a c, equals nought. But is it allowed? Multiplying an equation through can change what solves it — multiply by nought and every number in the world becomes a solution.

3:21So it was checked. Four hundred and fifty-five quadratics had their roots found on a grid, before and after. Not one root moved, and not one appeared. The same comparison for multiplication by nought went wrong all four hundred and fifty-five times, which is what makes the first result worth stating. And four a is not nought, because if a were nought there would be no quadratic here at all.

3:52Now look hard at the first two terms: four a squared x squared, plus four a b x. The first is the square of two a x, and the second is twice two a x times b — which is the beginning of a square you already know. Two a x plus b, all squared, is four a squared x squared, plus four a b x, plus b squared. That is our first two terms exactly, plus a b squared we never had. So we put it in and take it straight back out.

4:24The equation becomes: two a x plus b, all squared, minus b squared, plus four a c, equals nought. And that is where the four a came from. Without it the leading term is not a perfect square and none of this happens. Move the last two terms across: two a x plus b, all squared, equals b squared minus four a c. The discriminant has arrived on its own. Nobody put it there.

4:55Look at that line before you do anything to it. The left-hand side is a square, and squares of real numbers are never negative. So if x is going to be real, the right-hand side cannot be negative either. That is the condition — not a rule bolted onto the formula, but a consequence of the left side being a square. Now finish. Take the square root of both sides: two a x plus b equals plus or minus the square root of b squared minus four a c.

5:30There is your plus-or-minus. It arrives from the square root and from nowhere else. Take the b across, divide by two a — the whole of it, both terms — and you have the formula. That derivation was checked two ways that share no working, and they agreed everywhere. The same test with the four a left off failed ten thousand nine hundred and seventy-four times, which is how you know the agreement was not free.

6:01The formula is more readable than it looks, if you split it in the right place: minus b over two a, plus or minus the root of the discriminant over two a. The first piece does not depend on the plus-or-minus at all. So the two roots are not scattered — they sit at equal distances either side of minus b over two a. Call that the centre. The root term is the spread.

6:29That gives you a free check on any pair of roots: if they are symmetric about a centre, they total twice it, which is minus b over a. Five and minus twelve total minus seven; there b is seven and a is one, so minus b over a is minus seven. It matches, in two seconds. Before any of this can be used, you have to get a, b and c out of the equation, and this is the step that costs people the most marks.

7:04Everything on one side, powers in order, and each coefficient carries its own sign. In x squared plus seven x minus sixty equals nought, c is minus sixty. Not sixty. Read it as sixty and the discriminant becomes forty-nine minus two hundred and forty — minus one hundred and ninety-one, which says there is no real root at all. Read it correctly and the discriminant is forty-nine plus two hundred and forty. Two hundred and eighty-nine.

7:38One sign, and the verdict inverts completely. And if the equation is not tidy — terms on both sides, or powers out of order — then a, b and c do not exist yet. Tidy first. Here is a situation that turns into a quadratic, worth doing slowly because of one step in the middle. A circular park is thirteen metres across, and two gates sit on its boundary at opposite ends of a straight line through the centre — so the gates are the full thirteen apart.

8:15A pole is to stand somewhere on the boundary, seven metres further from one gate than from the other. Is that possible, and if so, where? Call the distance from the near gate x. Then the far one is x plus seven. Now the step that does the real work. Any point on the boundary sees that line through the centre at a right angle. So the pole, the two gates and the two distances make a right-angled triangle, with the thirteen as its longest side.

8:49That is not something to take on faith. Four hundred and eighty-four points of the boundary were taken, and for every one the two distances squared totalled a hundred and sixty-nine — thirteen squared. A point just off the boundary failed the same test every single time. So the right angle is real, and the story is now an equation. Pythagoras gives it to us: x plus seven, all squared, plus x squared, equals a hundred and sixty-nine.

9:25Tidy that, and halve it — which, like the four a, moves no roots — and we have x squared plus seven x minus sixty equals nought. Before solving anything, answer the question that was actually asked. The discriminant is two hundred and eighty-nine, which is not negative, so yes: a position exists. Notice that we knew that before we knew where it was. Now the formula, with a one, b seven and c minus sixty: minus seven, plus or minus the root of two hundred and eighty-nine, over two.

10:04The root of two hundred and eighty-nine is seventeen. So x is five, or x is minus twelve. Both genuinely solve the equation. But x was a distance, and a distance cannot be minus twelve metres. So the pole stands five metres from one gate and twelve from the other. Twenty-five plus a hundred and forty-four is a hundred and sixty-nine, and twelve minus five is seven. One more case, where the plus-or-minus does nothing at all.

10:39Three x squared minus two x plus a third equals nought, so a is three, b is minus two and c is a third. The discriminant is four, minus four times three times a third. Four minus four. Nothing. So the spread is nothing and both roots land on the centre. Minus b over two a is two over six, which is a third. One number — though the honest way to say it is that the formula returned two roots which happen to be the same number.

11:14Check it. Three times a third squared is a third. Minus two times a third is minus two thirds. Plus a third. A third, minus two thirds, a third. Nothing. It passes. Every answer so far has been a whole number or a simple fraction. So what is the formula FOR? Here it is. Two x squared minus six x plus three equals nought. The discriminant is thirty-six minus twenty-four. Twelve.

11:47Twelve is not negative, so there are two real roots — but twelve is not a perfect square either. Root twelve is two root three, so the roots are six plus or minus two root three over four, which tidies to three plus or minus root three, over two. Those are irrational. No splitting of the middle term will ever reach them, because splitting looks for whole-number factors and there are none to find.

12:16The formula is not a faster way to factorise. It is what you use when there is no factorisation to find. And leave it in that form. A decimal is only an approximation of it. A word on how that derivation was tested, because if you expand the square and compare it against the same expansion, of course it agrees. That is not a check; it is an echo. So the two sides were built by routes that share no working. One expands into coefficients and never puts a number in; the other puts numbers into both sides and never expands anything.

13:00And running beside them, on the same line of the same test, was a third version with the four a deliberately left off — which fails, as it must. If it had ever stopped failing, the test would have stopped testing. The arithmetic with root three in it was done exactly. Root three is not a decimal, and no decimal appears anywhere in the checking. The formula is minus b, plus or minus the root of b squared minus four a c, all over two a — one bar, spanning both terms.

13:39It is only good for real roots while b squared minus four a c stays at nought or above, and that is part of the statement, not an afterthought. It comes from multiplying by four a and completing a square — four lines, and both the discriminant and the plus-or-minus fall out on the way. The plus-or-minus is the square root's doing, and it means two answers, not a choice.

14:07The roots sit either side of minus b over two a, so they total minus b over a — a check worth two seconds. And when a situation hands you the equation, the formula answers the equation — you still decide which root the situation can use.