Question Video: Finding an Unknown in a Quadratic Equation given the Ratio between Its Roots Mathematics

Video Transcript

The roots of the equation six 𝑥 squared minus 𝑚𝑥 plus 24 equals zero are positive numbers which are in the ratio four to nine, what is the value of 𝑚?

Well, first of all, what we have is a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. But why is this useful? Well, it’s useful because we have a couple of relationships to do with the roots. And these are that the sum of the roots is equal to negative 𝑏 over 𝑎 and the product of the roots is equal to 𝑐 over 𝑎. Well, what we’re gonna need to solve this problem is first of all take a look at our ratio. So we’ve got the ratio four to nine. So what we’re gonna think about it as is four 𝑥 to nine 𝑥. And this is because if we imagine 𝑥 to be the value of one part in our ratio. And therefore, what we can consider are our roots are four 𝑥 and nine 𝑥.

So, using this, what we can do now is use our first relationship. Well, in order to use this relationship, what we need to know is our 𝑎-, 𝑏-, and 𝑐-values, where they are 𝑎 is equal to six, 𝑏 is equal to negative 𝑚, and 𝑐 is equal to 24. So therefore, if we use our relationship, what we’re gonna get is four 𝑥 plus nine 𝑥, our roots, is equal to 𝑚 over six, which is gonna give us 13𝑥 is equal to 𝑚 over six. So then what we can do is multiply both sides of our equation by six. And when we do that, what we’re gonna get is 78𝑥 is equal to 𝑚.

Okay, great. But what we want to do now because we still want to find the value of 𝑚? Well, what we can do is use our second relationship to help us find out the value of 𝑥 and hence then find the value of 𝑚. While using the second relationship, what we’re gonna get is four 𝑥 multiplied by nine 𝑥, so the product, is equal to 24 over six, which is gonna give us 36𝑥 squared equals four. So then what we do is divide both sides of the equation by 36. And when we do this, what we get is 𝑥 squared equals four over 36. Well, then what we can do is take the square root of both sides of our equation. And when we do that, we get 𝑥 is equal to root four over root 36, because root four over 36 is the same as root four over root 36.

It’s worth noting at this point there we’re not interested in the negative value that we might get if we take the square root. And that’s because we’re told the roots are positive numbers. So then if we take the square root of four and the square root of 36, we’re gonna get 𝑥 is equal to two over six. So if we simplify this, we’ll get 𝑥 is equal to a third. Okay, great. So we now know the value of 𝑥. So what we can now do is substitute 𝑥 equals a third into our 78𝑥 equals 𝑚. And when we do this, we’re gonna get 78 multiplied by a third equals 𝑚. Well, 78 multiplied by a third is the same as 78 divided by three. Well, 78 divided by three is 26. So we’re gonna get 26 equals 𝑚.

So therefore, what we can say is that if the roots of the equation six 𝑥 squared minus 𝑚𝑥 plus 24 equals zero are positive numbers, which are in the ratio of four to nine, then the value of 𝑚 is 26.