The Graph and the Sign of a Quadratic Expression

Transcript

0:00Introduction

Hello everyone! Okay, one small challenge before we start. Take two x squared, minus three x, plus five. Is it positive for every real x? You could put x equals zero, then one, then two, and keep going. But there are infinitely many real numbers, right? You can never check them all. So how do we prove it? By the end of this video, you will answer it in one line. And remember, JEE asks this idea again and again, in many forms. So, let's begin.

0:37Complete the square: the graph is a parabola

So, first the picture. Let f of x be a x squared, plus b x, plus c. Complete the square, and it becomes: a, times, x plus b over two a, the whole squared, minus D over four a. Expand it at home and check, okay? Here capital D is our old friend, the discriminant: b squared minus four a c. Clear? So the graph of f is a parabola. Its vertex, the turning point, is minus b over two a, comma, minus D over four a. And its axis, the line of symmetry, is x equals minus b over two a. Now tell me. If a is positive, which way does this parabola open: up, or down? Look at the squared term. Upward, that's right! When a is positive, the squared term only grows as x moves away from the axis, so the curve goes up on both sides, like a cup. And when a is negative, it opens downward. Simple as that. And it meets the x axis exactly at the real roots. Remember this, because today the whole story is one question: where does the parabola sit, above the x axis or below it?

1:50Always positive, always negative

Now our hook question, in graph language. f of x positive for every x means the whole parabola sits above the x axis. Look at the three pictures: from the left, D greater than zero, D equal to zero, and D less than zero. All three have a positive. Which one stays above the axis everywhere? Yes, the last one, where D is less than zero! No real root, so it never touches the axis. So, f of x is greater than zero for all x, if and only if, a is greater than zero and D is less than zero. The double arrow means if and only if. Both conditions, okay? And what about greater than or equal to zero, the sign with a line below? Now touching the axis is allowed, like the middle picture. So a greater than zero, and D less than or equal to zero. Got it? Now flip it. When is f of x negative for every single x? Exactly! Turn the picture upside down. So a less than zero, opening downward, and D less than zero, never reaching the axis. But be careful: D less than zero alone does not tell you positive or negative. The sign of a decides that. Never forget a!

3:09Two real roots: outside same, inside opposite

Next case: D greater than zero. Now there are two real roots. Call them alpha and beta, these two Greek letters, with alpha the smaller one. Look at the first picture. Here a is positive, and the parabola cuts the axis at two points. So where is the curve below the x axis? Only between the two roots, exactly! Everywhere outside, it is above. So outside the closed interval alpha, beta, f of x has the sign of a. The square brackets mean both ends are included. And strictly between the roots, f of x has the opposite sign. Remember it like this: outside, same as a; inside, opposite. Two words, and the sign is done! And if D equals zero? The middle picture: it just touches the axis. So f of x has the sign of a everywhere, except at the double root, where it is zero, okay?

4:06The JEE trap: a parameter in the x² coefficient

Now, a classic trap in JEE problems. Suppose the coefficient of x squared is not a fixed number. It contains a parameter, like k or m. What happens if that coefficient becomes zero? Is it still a parabola? No! The x squared term vanishes, and the expression becomes linear, a straight line. And a linear expression is never of one sign for all x, unless it is just a constant. Because a slanting line always crosses the x axis, right? Positive on one side, negative on the other. So before you use a and D, check the case coefficient equals zero separately. Students skip this, and lose marks. Got it?

4:49Example 1: always positive test, sign between the roots

Now, remember our hook? Let's finish it. Show that two x squared, minus three x, plus five, is greater than zero for every real x. And also, find the sign of three x squared, minus seven x, plus two. Pause the video, and try it first. Done? Okay, let's check. For the first one, a is two, positive. And D is b squared minus four a c: minus three, squared, is nine, and four times two times five is forty. Nine minus forty: negative. So a is positive and D is negative. Always positive, that's it! Infinitely many values of x, and our hook is solved in one line. See? Now the second one. Three x squared, minus seven x, plus two, factorises as three x minus one, times x minus two. So the roots are one third and two, okay? Here a is three, positive. So tell me: between one third and two, what is the sign? Negative, of course! Opposite to a, because we are between the roots. The round brackets mean an open interval, so the end points are not included. And at the roots themselves, it is zero. And outside, for x less than one third or x greater than two, it has the same sign as a: positive. Clear?

6:18Example 2: which m makes it negative for all x

Now a harder one. For which values of m is this expression less than zero for all x: m minus one, times x squared, minus two m x, plus m plus two? Pause and try. Hint: remember the trap. Done? Okay. Most students jump straight in: a less than zero, and D less than zero, finished, right? No, no, careful! What must you check first? Yes, the coefficient of x squared, m minus one, can be zero! At m equals one, the expression is minus two x, plus three. Linear. And a linear expression is not always negative, so m equals one is out. Now m not equal to one; the slashed equals sign means not equal. For always negative, we need a less than zero, so m minus one less than zero, and D by four less than zero. D by four is D divided by four, so it has the same sign as D, and here it is easier to work with. So, D by four equals m squared, minus, m minus one times m plus two. Expand it: the m squared terms cancel, and we are left with just two minus m. Nice and clean! The first condition gives m less than one. And two minus m less than zero gives m greater than two. Now, can m be less than one and greater than two, at the same time? No way! Impossible. So there is no such m at all. Surprising, right? And notice, we checked m equals one first. Make that a habit, okay?

7:58Quick revision: a and D decide the sign, check a = 0 first

So, quick revision. Always positive: a greater than zero and D less than zero. Always negative: a less than zero and D less than zero. Two real roots: the sign of a outside, the opposite sign strictly between. And a parameter in the x squared coefficient? Check coefficient equals zero first. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!