The shape an equation has to have before it counts as quadratic

0:00A trust wants to carpet the floor of a hall. The carpet is to cover three hundred square metres, and the hall is to be one metre longer than twice its breadth. One number is unknown, so give it a name. Let the breadth be x metres. Then the length is not a second unknown. It is two x plus one. That is the modelling step, and it is the only place any thinking is required.

0:28The area is the breadth times the length: x times, bracket, two x plus one. Multiply it out and you get two x squared plus x. So the condition is two x squared plus x equals three hundred. Move the three hundred across and everything is on one side: two x squared plus x minus three hundred equals nought. That equation has a name, and the whole of this video is about what the name means.

1:01You have met the expression before. A quadratic polynomial is anything of the form a x squared plus b x plus c, with a not nought. Two x squared plus x minus three hundred is exactly that, with a equal to two, b equal to one and c equal to minus three hundred. An expression is not an equation. It does not say anything until you set it equal to something.

1:30Set a quadratic polynomial equal to nought, and what you have is a quadratic equation. That is the whole definition, and everything else in this video is learning to apply it honestly. The standard form is a x squared plus b x plus c equals nought, and it pins down four separate things. One letter, and one only. An equation in two letters is a different kind of object. The coefficients a, b and c are real numbers. Not necessarily whole numbers — root two is a perfectly good coefficient, and so is an eighth.

2:13The highest power of the letter that survives is exactly two. Not at most two, and not at least two. Exactly two. And a, the number multiplying x squared, is not nought. Those last two conditions are the ones that do the work, and they are the two people forget. Notice also what is NOT required. b may be nought, and so may c. Minus x squared plus three hundred and one equals nought has no x term in it at all, and it is a perfectly good quadratic equation.

2:51Take that last condition seriously for a moment. Why can a not be nought? Because if it is, there is no x squared term. It is not that the equation becomes awkward. It is that it becomes a different kind of equation. Watch what happens to a x squared plus five x plus six equals nought as a runs through the values minus three up to three. For every value except one, you have a quadratic. For a equal to nought, the x squared term vanishes and five x plus six equals nought is left, which is linear.

3:30That was swept rather than asserted: seven values of a, six of them leaving degree two and exactly one leaving degree one. So the condition is not a technicality. It is what stops the definition from covering everything. Now, standard form is a form. Plenty of quadratic equations do not arrive in it. Four x minus three x squared plus two equals nought is quadratic. Its powers are simply written in the wrong order.

4:05Arrange them downwards and it reads minus three x squared plus four x plus two equals nought, so a is minus three, b is four and c is two. Notice that a is negative, and that is allowed. The condition is that a is not nought, not that a is positive. One minus x squared plus three hundred equals nought is quadratic too. Collect the constants and it is minus x squared plus three hundred and one equals nought.

4:39Here b is nought, which the definition permits, and c is three hundred and one, which it does not object to either. Rewriting in standard form is not tidiness. It is the only arrangement in which a, b and c can be read off at all, and every tool that comes later takes those three numbers as its input. Which brings us to the working order, and it is the whole point of the topic.

5:09You cannot tell whether an equation is quadratic by looking at it. You have to tidy it first, and then look. Expand every bracket. Collect the like powers. Move everything onto one side. Then, and only then, ask what the highest surviving power is. The word doing the work there is SURVIVING. An x squared that cancels is not there. An x cubed that cancels is not there either. Degree is a property of what is left, not of what was written down.

5:43There are exactly two ways to be caught out by this, and they point in opposite directions. Both are worth seeing. Here is the first. x times, bracket, x plus one, close bracket, plus eight, equals, bracket, x plus two, times, bracket, x minus two. There is an x squared on the left. There is an x squared on the right. Every instinct says quadratic. Tidy it. The left side is x squared plus x plus eight.

6:18The right side is a difference of two squares: x squared minus four. Now subtract. The x squared terms are identical, so they go. What is left is x plus twelve equals nought. Degree one. This is a linear equation, and it has exactly one answer, minus twelve. A glance said two. The tidying said one. The tidying is right, because the x squared did not survive. And here is the opposite mistake. Bracket, x plus two, close bracket, cubed, equals x cubed minus four.

6:57There is a cube on both sides, so surely this cannot be quadratic. Expand the left. x plus two, all cubed, is x cubed plus six x squared plus twelve x plus eight. Subtract the right side. The x cubed terms are identical, and they cancel exactly. Six x squared plus twelve x plus twelve equals nought. Degree two. It is quadratic, and the cube on each side had nothing to do with it.

7:29You may divide right through by six, because multiplying or dividing an equation by a number that is not nought moves no answer. x squared plus two x plus two equals nought. That was checked rather than assumed, over three equations and a grid of points, against a scaling by nought that moves plenty. Eight equations, and the job is to sort them. For each one, tidy first and judge second.

8:01The first tidies to x squared plus seven equals nought. Quadratic, and with no x term at all. The second gives x squared minus four x plus six. Quadratic. The third has an x squared on each side, and both of them go. Minus three x plus one is left. Not quadratic. The fourth gives x squared minus ten x minus three, and the fifth x squared minus eleven x plus eight. Both quadratic.

8:34In the sixth the squares cancel and seven x minus three remains. Not quadratic. Now the sharpest pair in the list, and they sit next to each other on purpose. Both of them start with a cube on each side. In the seventh the cubes do NOT cancel: one side has x cubed and the other has two x cubed. Minus x cubed plus six x squared plus fourteen x plus eight remains. Degree three, so not quadratic.

9:06In the eighth they DO cancel, and two x squared minus thirteen x plus nine is left. Quadratic. Five of the eight are quadratic. Two of them look quadratic and are not. And the only way to know which is which was to do the arithmetic. One last thing, because this shape of question is very old. The Babylonians could find two numbers from their sum and their product, which is exactly the equation x squared minus p x plus q equals nought.

9:43Ask for two numbers adding to seven and multiplying to twelve and you are asking for x squared minus seven x plus twelve to be nought. The answer is three and four. Euclid did the same problem as one about lengths, geometrically. Brahmagupta set down a rule for the type a x squared plus b x equals c, which is quadratic the moment you move the c across. Al-Khwarizmi worked through the several types systematically.

10:16Sridharacharya is credited with the formula reached by completing the square — the one this subject still uses. And Abraham bar Hiyya Ha-Nasi published complete solutions in Europe. Different places, different centuries, and the same shape of question every time. A word on the checking, because the claim here is unusually easy to confirm without testing anything. For most equations, what a glance sees and what survives agree, so a checker reading the degree off the written form would look right almost everywhere.

10:57So both questions were asked, of every equation, and the answers filed together as a pair. Nearly two thousand equations were built by putting products of small factors on each side. Each was asked what the higher of the two sides' degrees is, before anything cancels, and then what the degree is after everything is moved to one side and collected. The diagonal of that table is the equations that look like what they are, and it is well populated.

11:32But so are the two cells the topic exists for. Seven hundred and twelve equations show an x squared on both sides and are not quadratic. Forty-four show a cube on both sides and are. Both of those had to be full. A checker reading the degree off the written form empties them; so does one that stopped looking at the written form at all. The degree itself was found twice, by routes that share no arithmetic.

12:03One expands the expression and counts coefficients. The other never builds a polynomial at all: it evaluates the expression at nought, one, two, three and so on, and takes differences until they vanish, which is a fact about the function rather than about how it is written. They agreed on every equation. And beside them ran a third route that forgets the cross terms when it multiplies out, which disagreed on seven hundred and twelve of them — which is what makes the other two agreeing worth anything.

12:42A quadratic equation is a quadratic polynomial set equal to nought. In standard form: a x squared plus b x plus c equals nought, one letter, real coefficients, highest surviving power exactly two, and a not nought. b may be nought. c may be nought. a may not. The powers may arrive in any order, and a may be negative. Neither of those stops it being quadratic. And the test is never a glance. Expand, collect, move everything to one side, and then ask what is left.

13:21An x squared on both sides can vanish, and then it was never quadratic. A cube on both sides can vanish, and then it was. Degree is what survives.