Question Video: Finding the Value of an Unknown in a Quadratic Equation given One of Its Roots Mathematics

Given that π‘₯ = βˆ’9 is a root of the equation π‘₯Β² + π‘šπ‘₯ = 36, determine the value of π‘š.

03:21

Video Transcript

Given that π‘₯ equals negative nine is a root of the equation π‘₯ squared plus π‘šπ‘₯ equals 36, determine the value of π‘š.

Well, the first thing we’re gonna do is we’re gonna look at π‘₯ squared plus π‘šπ‘₯ 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 π‘₯ squared plus π‘šπ‘₯ minus 36 is equal to zero. So what we’re told in the question is that π‘₯ 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 π‘₯ plus nine multiplied by π‘₯ plus π‘Ž, 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 π‘₯ plus nine? Well, we know that if π‘₯ is equal to negative nine is a root, then when we’re writing a quadratic in factored form, the value of π‘₯ is going to be what’s required to make each of the parentheses equal to zero. So if π‘₯ 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 π‘Ž? Because once we found our π‘Ž, then we’re gonna be able to find the value of π‘š.

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 π‘₯ multiplied by π‘₯, which is π‘₯ squared, then π‘₯ multiplied by π‘Ž, which is π‘Žπ‘₯, then positive nine multiplied by π‘₯, which is nine π‘₯, and then finally nine multiplied by π‘Ž, which is nine π‘Ž. 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 π‘₯ squared plus π‘šπ‘₯ minus 36 equals zero. So what we can see is that negative 36 is the only non-π‘₯-containing term. So if we look back at the equation that we form now, what we’ve got is positive nine π‘Ž is the only non-π‘₯-containing term.

So therefore, what we can do is equate these. So we can say that nine π‘Ž is equal to negative 36. So therefore, what we can do is divide through by nine to find out what π‘Ž is. And when we do that, we’re gonna get π‘Ž is equal to negative four. So now what we’re gonna do is substitute this value of π‘Ž back in to our equation. So when we do, we’ll get π‘₯ squared minus four π‘₯ plus nine π‘₯ minus 36 is equal to zero. So now what we can do is collect like terms, so we can collect negative four π‘₯ plus nine π‘₯. So what we’re gonna have is π‘₯ squared plus five π‘₯ minus 36 is equal to zero.

Well, if we look at the question, what we’re looking to do is determine the value of π‘š. Well, we can see from our new equation that we’ve found the value of π‘š. And that value of π‘š 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 π‘₯ is equal to negative nine is a root of the equation π‘₯ squared plus π‘šπ‘₯ equals 36, then the value of π‘š is five.

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