Video Transcript
Given that 𝑥 equals negative nine is a root of the equation 𝑥 squared plus 𝑚𝑥 equals 36, determine the value of 𝑚.
Well, the first thing we’re gonna do is we’re gonna look at 𝑥 squared plus 𝑚𝑥 equals 36. And we can see that, in fact, what we can do with this is rewrite it in a quadratic equation form. So if we subtract 36 from each side of the equation, what we’ll do is see our quadratic in a form we’re more used to. So we’ve got 𝑥 squared plus 𝑚𝑥 minus 36 is equal to zero. So what we’re told in the question is that 𝑥 is equal to negative nine is a root of the equation. So therefore, what we can do is rewrite our quadratic equation in factored form. And when we do that, we’re gonna have 𝑥 plus nine multiplied by 𝑥 plus 𝑎, so some number, is equal to zero. And that’s because we only know one of the roots of our equation.
You might think, well, how do we know that the left-hand parentheses in our factored form was 𝑥 plus nine? Well, we know that if 𝑥 is equal to negative nine is a root, then when we’re writing a quadratic in factored form, the value of 𝑥 is going to be what’s required to make each of the parentheses equal to zero. So if 𝑥 is equal to negative nine, then negative nine plus nine is equal to zero. Okay, great. But how are we gonna find out our mystery number, our 𝑎? Because once we found our 𝑎, then we’re gonna be able to find the value of 𝑚.
So what we’re gonna do is distribute across our parentheses. And to do this, what we do is multiply everything in the left-hand parentheses by everything in the right-hand parenthesis, which is gonna be first of all 𝑥 multiplied by 𝑥, which is 𝑥 squared, then 𝑥 multiplied by 𝑎, which is 𝑎𝑥, then positive nine multiplied by 𝑥, which is nine 𝑥, and then finally nine multiplied by 𝑎, which is nine 𝑎. So we’ve got this all equal to zero. Okay, so now what’s our next step? Well, if we take a look at the original equation, so our original quadratic, we’ve got 𝑥 squared plus 𝑚𝑥 minus 36 equals zero. So what we can see is that negative 36 is the only non-𝑥-containing term. So if we look back at the equation that we form now, what we’ve got is positive nine 𝑎 is the only non-𝑥-containing term.
So therefore, what we can do is equate these. So we can say that nine 𝑎 is equal to negative 36. So therefore, what we can do is divide through by nine to find out what 𝑎 is. And when we do that, we’re gonna get 𝑎 is equal to negative four. So now what we’re gonna do is substitute this value of 𝑎 back in to our equation. So when we do, we’ll get 𝑥 squared minus four 𝑥 plus nine 𝑥 minus 36 is equal to zero. So now what we can do is collect like terms, so we can collect negative four 𝑥 plus nine 𝑥. So what we’re gonna have is 𝑥 squared plus five 𝑥 minus 36 is equal to zero.
Well, if we look at the question, what we’re looking to do is determine the value of 𝑚. Well, we can see from our new equation that we’ve found the value of 𝑚. And that value of 𝑚 is five, remembering that we need to make sure that we do take into account the sign, and it is positive. So we can say that given that 𝑥 is equal to negative nine is a root of the equation 𝑥 squared plus 𝑚𝑥 equals 36, then the value of 𝑚 is five.
