What the sum and product of two zeroes reveal about a, b and c

0:00Here is a quadratic: two x squared minus eight x plus six. Split the middle term into minus six x and minus two x, group the pairs, and it factors as two times the bracket x minus one times the bracket x minus three. So its zeroes are one and three. Now do something the factoring never asked you to do. Add the two zeroes together. One plus three is four.

0:29And look back at the polynomial. b is minus eight, a is two, and eight over two is four. The same number. Now multiply the two zeroes instead. One times three is three. c is six, a is two, and six over two is three. The same number again. Two numbers you had to do real work to find, sitting in plain sight in the coefficients you started with. One example is an accident.

1:01Take another: three x squared plus five x minus two. Multiply a by c to get minus six, split the middle term into six x and minus x, and it factors as the bracket three x minus one times the bracket x plus two. The zeroes are one third and minus two. Add them and you get minus five thirds. And minus b over a is minus five over three.

1:29The same. Multiply them and you get minus two thirds. And c over a is minus two over three. The same. Two examples is still not a reason. But notice what changed and what did not. The leading coefficient moved from two to three, the zeroes stopped being whole numbers, and the pattern did not care. That is the shape of something that is true for a reason, and the reason is worth more than either example.

2:03So stop looking at quadratics and start building one. Suppose I tell you the zeroes are alpha and beta, and nothing else. You can write down a polynomial immediately. The bracket x minus alpha is nought when x is alpha. The bracket x minus beta is nought when x is beta. Multiply them and you have something that is nought at both. But that is not the only answer, and here is the step almost everybody skips.

2:34Double the whole thing and it is still nought at both. Multiply it by minus seven and it is still nought at both. Multiplying by any number that is not nought moves neither zero. So the honest thing to write is k times the bracket x minus alpha times the bracket x minus beta, with k standing for a number nobody has told us yet. Now expand it, slowly, because every term is about to earn its keep.

3:07The two brackets multiply out to x squared, minus alpha x, minus beta x, plus alpha beta. Collect the two middle terms and that is x squared minus the bracket alpha plus beta, times x, plus alpha beta. Then multiply the whole line by k. That gives k x squared, minus k times the bracket alpha plus beta, times x, plus k alpha beta. Read the three coefficients off that line.

3:40The x squared coefficient is k. The x coefficient is minus k times the sum of the zeroes. The constant is k times the product of the zeroes. Every one of those was checked against the multiplication itself, for six hundred and forty eight different choices of k, alpha and beta. And a quadratic is also written a x squared plus b x plus c. Two ways of writing the same object, so the coefficients have to agree, one at a time.

4:16The x squared coefficient gives a equals k. That single line is the whole point, and it is the one that gets skipped. The mystery number out in front is not some extra thing. It is the leading coefficient, the a you were already looking at. The x coefficient gives b equals minus k times the sum. The constant gives c equals k times the product. And since k is a, we can replace it: b is minus a times the sum, and c is a times the product.

4:52Two lines to finish. b equals minus a times the sum, so divide both sides by a and the sum is minus b over a. c equals a times the product, so divide both sides by a and the product is c over a. There they are. The sum of the zeroes is minus b over a. The product of the zeroes is c over a. And now you can see why they are ratios rather than statements about b and c on their own.

5:26The k had to be divided out, and dividing by k is dividing by a. Anyone who remembers the sum as minus b has silently assumed that a is one. That minus sign is the only thing here you can get wrong quietly. Take x squared plus seven x plus ten. Its zeroes are minus two and minus five. Multiply them: two negatives make a positive, so the product is ten, which is c over a, and everything looks fine.

6:01Add them and you get minus seven. Not seven. The b in that polynomial is plus seven, and the zeroes add to its opposite. The reason the mistake survives so long is that it hides. Whenever b is nought, plus b over a and minus b over a are the same number, and a wrong sign gives the right answer. So the checking behind this video did not just confirm the relation.

6:30It counted the quadratics that can tell the two signs apart — four hundred and thirty two of them — and required the wrong sign to fail on every single one. Now a case that catches people out. Four s squared minus four s plus one is a perfect square: the bracket two s minus one, all squared. It has one zero, at one half. The curve comes down, touches the axis, and turns back without ever crossing.

7:03So what does the sum of the zeroes mean when there is only one? Check what the relation demands: minus b over a is four over four, which is one. And one half is not one. The relation has not broken. The bracket appears twice, so the zero is used twice, and you add it twice. One half plus one half is one. The product is one half times one half, a quarter, and c over a is one over four.

7:33A repeated zero is one number doing two jobs, and both relations count the jobs, not the numbers. The relations also do not care whether the zeroes are the sort of number you can write down neatly. Take x squared minus three. Its zeroes are root three and minus root three, and neither of those is a ratio of whole numbers — three is not a whole number squared, so its square root cannot be one.

8:06Add them and the roots cancel exactly: the sum is nought, and minus b over a is nought over one. Multiply them and you get minus root three times root three, which is minus three, and c over a is minus three over one. Two awkward numbers went in and two perfectly ordinary ones came out. That is worth noticing, because it is the direction the relations are useful in.

8:36They let you say something exact about zeroes you cannot write down. One trap before we turn the whole thing round. The relations read b and c off the polynomial, so you have to know which number is which. Here is six x squared minus three minus seven x. The terms are out of order. The number sitting next to the x is minus seven, so b is minus seven, and c, the term with no x at all, is minus three.

9:08Anyone who reads left to right and takes b as minus three gets both relations wrong at once. Put them in order first. Its zeroes are minus a third and three halves, they add to seven sixths, and minus b over a is seven over six. Everything so far has run one way: zeroes in, coefficients out. Turn it round. Suppose I want a quadratic whose zeroes add to minus three and multiply to two.

9:42The sum is minus b over a, so minus b over a is minus three. The product is c over a, so c over a is two. The simplest way to satisfy both is to take a as one, which makes b three and c two. So x squared plus three x plus two. Check it: that factors as the bracket x plus one times the bracket x plus two, its zeroes are minus one and minus two, and those do add to minus three and multiply to two.

10:17So a sum and a product are enough to build a polynomial. Enough to build a polynomial — but not enough to pin down which one. Go back to the k. The sum is minus b over a and the product is c over a, and both of those are ratios. Multiply a, b and c all by the same number and neither ratio moves. So two x squared plus six x plus four has the same sum and the same product.

10:49So does three x squared plus nine x plus six. So does the one you get from a half, and from minus two. Eight members of that family were built and checked here: every one of them has the same two zeroes, minus one and minus two, every one has the same sum and product, and no two of them are the same polynomial. Drawn together they are eight different curves, some steep, some shallow, some opening downwards, all passing through the axis at the very same two points.

11:25A sum and a product name a family, and choosing a is choosing which member you meant. There is a sharper limit, and it is easy to walk straight past. Building coefficients from a sum and a product always works. Whatever two numbers you hand me, I can write the quadratic down. But that does not mean any real number is a zero of it. Ask for a sum of nought and a product of root five and you get x squared plus root five, which is never nought for any real x at all.

12:00Ask for a sum of one and a product of one and you get x squared minus x plus one, which has no real zeroes either. Ask for minus a quarter and a quarter: the same story. Six such requests were tested here, exactly three of which describe no real zeroes at all, and the sign that decides it was worked out in exact arithmetic rather than in decimals, because the numbers involved are square roots.

12:33So the construction is a statement about coefficients. It is not a promise that the zeroes you asked for exist. So: the sum of the zeroes is minus b over a, and their product is c over a. Not a pattern noticed in examples, but a consequence of writing a quadratic as k times two brackets and matching the coefficients — which is where a turns out to be k, and why both relations are ratios.

13:06The minus sign belongs to the sum alone. A repeated zero is counted twice. Irrational zeroes obey it exactly, which is what makes it useful. And run backwards it gives you a family of polynomials rather than one, with no guarantee that the zeroes are real. Everything counted here was checked before a word was written. Seven hundred and twenty six quadratics were measured by three methods that share no reasoning, and the one doing the refereeing never used a formula at all: it hunted for sign changes along a grid and squeezed each zero into a bracket narrower than a millionth, so the numbers it handed back had never seen b or c.