CHAPTER 03.01: QUADRATIC EQUATIONS: The Quadratic Equation Formula
In this segment, we're going to talk about the formula for a quadratic equation. So, let's go ahead and see what a typical quadratic equation looks like. So generally it will look like this, a x squared plus b x plus c equal to 0. And the formula for finding the solution for it, for a not equal to 0 is as follows, that x is equal to -b plus minus square root of b squared minus 4 a c, divided by 2 a. It basically means that there are two roots, one root being -b plus square root of b squared minus 4 a c, divided by 2 a, and the other root is -b minus square root of b squared minus 4 a c, divided by 2 a. So those are the two roots of the . . . of a typical quadratic equation, and once you find out what these . . . these values are, you'll be able to find out what the two roots of the equation are. A thing to note is that this is called the discriminant, so square root of b squared minus 4 a c is called the discriminant. And based on whether . . . what the value of this b squared minus 4 a c is, your roots can take certain . . . certain forms. So what does that mean? That means as follows, that your . . . if your b squared minus 4 a c is greater than 0, the roots are real. The reason why the roots are real is because b squared minus 4 a c is greater than 0, when you take the square root of b squared minus 4 a c, we get a positive number, and a negative number, so it doesn't matter, you take the positive root of it, because you have plus minus in front of that, so if b squared minus 4 a c was greater than 0, then the roots are real. If b squared minus 4 a c is equal to 0, then the roots are . . . roots are identical. And the reason why the roots are identical is because when b squared minus 4 a c is equal to zero, you don't have the plus minus part of it, so the roots will be identical for that case. Now, if b squared minus 4 a c is less than 0, then the roots are complex. And the reason why the roots are complex is because if b squared minus 4 a c, the square root of b squared minus 4 a c will be an imaginary number, hence you will get complex roots for that. So let me, again, explain, if you look at x is equal to the . . . the roots of the typical quadratic equation look like this, b squared minus 4 a c divided by 2 a. So if b squared minus 4 a c is a positive number, then . . . then the square root of that is taken as the positive number, you can take as the negative number and that's not going to affect anything so far as the roots are concerned, because you have a plus and a minus here, so if b squared minus 4 a c is greater than 0, you take the positive square root of that, and then you can see that you are going to get real roots and distinct roots of the quadratic equation. If b squared minus 4 a c is equal to 0, then this quantity here is 0, so you will have -b divided by 2 a as one root, and -b by 2 a is the other root, so the roots are identical, or what people might call as the roots are repeated. b squared minus 4 a c is less than 0, then you can see that this quantity here becomes a negative quantity, so the square root of a negative quantity will be an imaginary number, so you will get complex roots of this particular quadratic equation, they'll be complex conjugates of each other, because you will have plus the imaginary number and negative of the imaginary number in your roots itself. And that's the end of this segment. |