Video Transcript
Determine whether the roots of the equation 𝑥 squared minus the square root of five 𝑥 minus one equals zero are rational or not without solving it.
We begin by noticing that the equation 𝑥 squared minus the square root of five 𝑥 minus one equals zero is a quadratic equation. In other words, it’s of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎 is not equal to zero. Since we have a quadratic equation, there are a number of ways we can determine its roots. Where possible, we can factor the quadratic part and then deal with the individual factors separately. In this case though, we’re not looking to solve the equation, just determine whether its roots are rational. And so let’s consider the quadratic formula. This says that the roots of the equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero are given by negative 𝑏 plus or minus root 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.
So there’s a couple of things going on here. First, we have the discriminant. Now this is the part inside the square root, 𝑏 squared minus four 𝑎𝑐. It follows if the expression 𝑏 squared minus four 𝑎𝑐 is not a square number, then the square root of the discriminant is going to be irrational. Similarly, if either 𝑏 or two 𝑎 are themselves irrational, then we are either dividing an irrational number by a rational number or vice versa. In these cases, we will also end up with an irrational result. Now of course, if we divide a rational number by a second rational number, then we have a rational number. If we divide two irrational numbers, it very much depends on what the irrational numbers are as to our result.
So let’s begin by looking at each individual part of the quadratic equation. 𝑎 is the coefficient of 𝑥 squared; it’s one. 𝑏 is the coefficient of 𝑥, so it’s negative root five. And 𝑐 is the constant negative one. So the discriminant 𝑏 squared minus four 𝑎𝑐 is negative root five squared minus four times one times negative one. Negative root five squared is positive five. Then when we subtract four times one times negative one, that’s the same as adding four. This means the discriminant 𝑏 squared minus four 𝑎𝑐 is nine. Nine is a square number, so the square root of nine is going to be a rational number. It’s three.
However, we said that if 𝑏 is irrational or 𝑎 is irrational, then the solutions are also irrational. Now 𝑏 is the square root of negative five. Since five is not a square number, root five, and hence negative root five, is not a rational number. 𝑎 is one, so two 𝑎 is also rational, meaning the denominator of our result is going to be rational. So our solutions are given by an irrational number plus or minus a rational number divided by a rational number. That gives an irrational result. And so we’ve determined that the roots of the equation 𝑥 squared minus root five 𝑥 minus one equals zero are not rational.
