### Video Transcript

The roots of the equation ππ₯ squared minus 12ππ₯ plus π equals zero, where π is not equal to zero, are πΏ and π. Given that πΏ is greater than π and πΏ minus π equals 20, does πΏ equal 10 plus six π over π?

Okay, so in this question, what we have is a quadratic equation in the form ππ₯ squared plus ππ₯ plus π equals zero. And what we have with a quadratic in this form is a couple of relationships to deal with the roots. And that is, first of all, that the sum of the roots is equal to negative π over π and the product of the roots is equal to π over π. Well, the first thing weβre gonna do before we use either of our relationships is take a look at πΏ minus π equals 20.

Well, the first thing we want to do is rearrange our equation so that either πΏ or π are the subject. So what weβre going to do is rearrange it so that π is the subject because what we want to do eventually is get everything in terms of πΏ. So, to enable us to do that, what weβre gonna do is add π and subtract 20 from each side of the equation. So when we do that, we get πΏ minus 20 equals π. So, great, weβve made π the subject. So now, whatβs the next step?

Well, now what we can do is use our first relationship and that is that the sum of the roots is equal to negative π over π. But to enable us to use this, the first thing we need to do is work out what the π, π, and π are from our quadratic. Well, our π, π, and π are π, negative 12π, and π. Well, if we substitute these values into our relationship for the sum of the roots, we can say that πΏ plus π, so the sum of our roots, is equal to 12π over π. We get 12π as the numerator because itβs negative π over π and we had π being equal to negative 12π. So, negative negative 12π gives us positive 12π.

So now, what weβre gonna do is substitute π in for our π because we know that π is equal to πΏ minus 20. So when we do this, what weβre gonna get is πΏ plus πΏ minus 20 equals 12π over π, which is gonna give us two πΏ minus 20 equals 12π over π. Well, then, if we add 20 to both sides, what weβre gonna get is two πΏ equals 12π over π plus 20. Well, great. Is this what weβre looking for? Is this the answer weβre looking to match? Well, no, because if we look at what weβve got, weβre looking for πΏ equals, but here we got two πΏ equals.

So we do need another step. And that step is to divide through by two. And if we divide each term by two, weβre gonna get πΏ equals 12π over two π plus 10. So if we take a look at the term which is a fraction, we can in fact divide through by two because two is a factor of 12 and two. So when we do this, what weβre gonna be left with is πΏ equals six π over π plus 10. Well, this is exactly what weβre looking for. So, therefore, we can say that yes, πΏ does equal 10 plus six π over π.