Why the sign of b² − 4ac settles how many real roots exist

0:00Here is a question about an equation, and it is not the question you are used to. Not 'what are the roots'. Just: how many are there? It sounds like the smaller half of the same job. It is not. You can answer it without solving anything, from one number you can work out in a single line. And that number is not really a number. What matters about it is only its sign.

0:28To see why, take the formula and split it in the right place. x is minus b over two a, plus or minus the square root of b squared minus four a c, over two a. The first piece has no plus-or-minus in it. It is one number, and both roots carry it. So think of it as a place: both roots start there. Call it the centre. The second piece is what tells them apart. One root moves that far to the right, the other that far to the left.

1:02That is a push. And the centre is genuinely a centre — the whole expression is symmetric about it. That was measured, not asserted. Over seventeen thousand offsets, the value a step right of the centre matched the value a step left, every single time. And the same test about a different point — minus b over a — failed on sixteen thousand of them. So the centre is that value and not another.

1:34Now look at where the count of answers could possibly change. Not in the centre. The centre is one number whatever a, b and c are, and it is there in both roots. It has to be the push. That is the only part that can behave differently. And there are exactly three things a push can be. It can be a real distance, in which case the two roots go to different places.

2:02It can be no distance at all, in which case they both stay where they started. Or there can be no real number available to push by — and then nothing real comes out at all. Three cases — one picture with the push set to three different things. Take the first case. b squared minus four a c is positive. Then its square root is a real number, and it is not nothing. So the push is a genuine distance.

2:33One root lands to the left of the centre, the other the same distance to the right, and they are two different numbers. Two real roots, and the word for them is DISTINCT — different from each other. That is the case you have met most often — which is why it is worth naming as a case at all. Now let b squared minus four a c be nothing at all.

2:59The square root of nothing is nothing. So the push has no length. Add nothing to the centre and you get the centre. Subtract nothing and you get the centre again. Both roots land on minus b over two a. One value, arrived at twice. And the careful way to say it is that there are two roots which happen to be the same number. The word for that is COINCIDENT — they coincide.

3:27That is not pedantry. The second one is hiding in a factorisation, and we will go and find it. The third case. b squared minus four a c is negative. Now the formula asks for the square root of a negative number, and there is no real number that answers. Because a square root is a number you multiply by itself. And a real number multiplied by itself is never negative.

3:57Positive times positive is positive. Negative times negative is also positive. And nothing times nothing is nothing. That was checked over twelve hundred and one numbers, six hundred of them negative. Not one square came out negative. So there is no real distance to push by, and no real root exists. Not hidden, not hard to find — there is nothing to find. The quantity doing all this work has a name. It is called the discriminant, and it is b squared minus four a c.

4:36And the name is exact. To discriminate is to tell things apart, and that is precisely what it does: it tells the three cases apart, and it does nothing else. Positive: two distinct real roots. Nothing: two real roots which coincide. Negative: no real roots. You will see both wordings for the middle one; learn both. And notice what is NOT in that list. Nothing about where the roots are, and nothing about how big anything is.

5:10Which brings up the mistake this topic actually produces: that a bigger discriminant means more roots. It does not. Take x squared minus one. Its discriminant is four, and its roots are one and minus one. Now take x squared minus ten thousand. Its discriminant is forty thousand — ten thousand times as large — and its roots are a hundred and minus a hundred. Two roots in both. The bigger discriminant did not buy a third one.

5:45What it bought was distance. The roots are further apart, because the push is longer. That was swept as well. Five thousand quadratics, sorted by the SIZE of their discriminant instead of its sign. The band of discriminants a hundred or more holds fifteen hundred equations with two roots and two hundred and forty with none. The size tells you nothing about the count. Sort the same five thousand by sign instead and every single one falls into line. Only the sign decides.

6:22So here is the whole method, on a real equation. Two x squared minus four x plus three equals nought. a is two, b is minus four, c is three. b squared. Careful here — b is minus four, and minus four squared is sixteen, not minus sixteen. Four a c is four times two times three, which is twenty-four. Sixteen minus twenty-four is minus eight. Negative. So there are no real roots, and we are finished. One line, no solving, no formula.

6:59And that sign trap is not a small thing. Read b squared as minus sixteen and you would get minus forty, which is still negative — but on other equations that slip flips the verdict outright. The real power of this shows up when the question is not about an equation at all. A pole is to stand on the boundary of a circular park thirteen metres across, seven metres further from one gate than from the other, the two gates sitting at opposite ends of a line through the centre.

7:36That situation turns into x squared plus seven x minus sixty equals nought. Now: is such a position possible? The discriminant is forty-nine minus four times one times minus sixty. And c is minus sixty, so that is minus two hundred and forty — which we are subtracting. Forty-nine plus two hundred and forty. Two hundred and eighty-nine. Positive. So yes. A position exists. And notice that we now know it exists without having the faintest idea where it is.

8:14Only afterwards do we find it: five metres from one gate, twelve from the other. Existence first, location second. They are different questions and the discriminant answers only the first. Now the promised second root, in the case where the discriminant is nothing. Take three x squared minus two root six x plus two equals nought. Its discriminant is twenty-four minus twenty-four. Nothing. And if you factorise that expression, you get the same linear factor twice.

8:52That is where the second root is. The factor appears twice, so it supplies its number twice. A vanishing discriminant and a repeated factor are not two facts. They are one event, seen from two directions. That was tested across the whole sweep, and tested the hard way. For every one of five thousand quadratics, a search hunted for a number r such that a times x minus r, all squared, IS the expression.

9:24The search knows nothing about discriminants. It just tries candidates. It found exactly forty-eight, and those forty-eight are exactly the forty-eight whose discriminant is nothing. Not one more, not one fewer. Because the discriminant is a single expression, you can also run the question backwards. Take two x squared plus k x plus three equals nought, and ask what k makes the roots equal. Equal roots means the discriminant is nothing. k squared minus twenty-four equals nought.

10:02So k is two root six, or minus two root six — both of them answers. Now a second one, and it has a trap in it. k x times x minus two, plus six, equals nought. Tidy it first: k x squared minus two k x plus six equals nought. Its discriminant is four k squared minus twenty-four k, which is four k times k minus six. That is nothing when k is nought, and when k is six.

10:35But look at k equals nought. There is no x squared term left at all. It is not a quadratic. And what remains is the bare statement that six is nothing, which is false — no value of x makes it true. So k equals six is the only answer, and six x squared minus twelve x plus six really is six times x minus one, squared. Three questions now, one of each case, and all three are answered by a discriminant alone.

11:10First: a rectangular grove, twice as long as it is broad, of area eight hundred square metres. Possible? With breadth x the equation is two x squared minus eight hundred equals nought, discriminant six thousand four hundred. Positive. So yes. And a hunt through fourteen thousand four hundred rectangles turns up exactly one: twenty metres by forty. Second: two friends whose ages total twenty, and whose ages four years ago multiplied to forty-eight.

11:47That gives x squared minus twenty x plus a hundred and twelve. Discriminant four hundred minus four hundred and forty-eight. Minus forty-eight. Negative. So the situation cannot occur. That is the answer — not a failure to find one, and the hunt through every candidate pair confirms it. Third: a rectangular park with eighty metres of boundary and an area of four hundred. Discriminant sixteen hundred minus sixteen hundred. Nothing.

12:22So it is possible, and possible in exactly one way — and that one way turns out to be a twenty-metre square. A word on the checking, because this claim is the easiest in the subject to confirm without testing anything. It is TRUE that the sign settles the count. So a checker that counts the roots using the formula — which has b squared minus four a c under its square-root sign — has proved precisely nothing.

12:55So the counting was done somewhere the discriminant is not. Each expression was evaluated at every twenty-fourth from minus sixteen to sixteen, and the roots were counted as the places where the value changed sign or landed on nothing. Three million eight hundred and forty-five thousand points, and nowhere in that walk is b squared minus four a c formed, or two a, or a square root. The two verdicts agreed on all five thousand: three thousand five hundred and sixty-four with two roots, forty-eight with one, one thousand three hundred and eighty-eight with none.

13:37And beside it the same table built from a mis-stated discriminant, b squared PLUS four a c, which gets the count wrong two thousand eight hundred and fifty-six times. That is what makes the agreement mean something. A table that is diagonal with nothing beside it is a table of one column. Both roots sit at minus b over two a, and are then pushed apart by the square root of b squared minus four a c, over two a.

14:10The push is the only part that can change the count, so the only question is what the push is. Positive discriminant: a real push, two distinct roots. Nothing: no push, two roots which coincide on one number — and a repeated factor is that same event seen from the other side. Negative: no real number to push by, and no real root. Only the sign decides. The size tells you how far apart they are, and nothing whatever about how many.

14:45Which is why 'is this possible?' is a question you can answer in one line, and answer honestly with a no.