Question Video: Finding Two Unknowns in a Quadratic Equation Using the Relation between Its Coefficients and Roots

Video Transcript

Given that one and 12 are the roots of the equation 𝑥 squared plus 𝑚𝑥 plus 𝑛 equals zero, find the values of 𝑚 and 𝑛.

So, what we’re told in this question is that one and 12 are the roots or solutions of the equation. And the equation we’re looking at is a quadratic in the form 𝑥 squared plus 𝑚𝑥 plus 𝑛 equals zero. But in fact, we have a couple of methods we can use to help us solve this problem. So, we’re gonna look at the first method, and the first method is gonna be involving the factored form of our quadratic.

Well, what we know is our quadratic in factored form is gonna be 𝑥 minus one multiplied by 𝑥 minus 12 equals zero. So, you might wonder, “Well, how did we get the factored form straightaway?” Well, when we have the factored form of a quadratic, the values of 𝑥, or our solutions or roots, are the values that make each of the parentheses equal to zero. So, for example, if you have 𝑥 equals one, then one minus one equals zero for the left-hand parentheses. And if you got 12 for our 𝑥-value, then 12 minus 12 equals zero. And I want one of our parentheses to be equal to zero because on the right-hand side of the equation, the result is zero. And if you have zero multiplied by anything, it will give us our zero. Okay, great. So, we’ve got our quadratic in factored form. So, what do we want to do next?

Well, now, to enable us to calculate what 𝑚 and 𝑛 are, what we’re gonna do is distribute across our parentheses. So first, what we’re gonna have is 𝑥 multiplied by 𝑥, which is 𝑥 squared. So then, what we’re gonna have is 𝑥 multiplied by negative 12, which gives us minus 12𝑥. And then, what we’re gonna move on to now is the negative one in the first parentheses. And that’s because we’ve multiplied the 𝑥 by both terms in the second parentheses. So, we have negative one multiplied by 𝑥. So, we have minus 𝑥. And then finally, we’ve got negative one multiplied by negative 12, which gives us positive 12. So, we’ve got 𝑥 squared minus 12𝑥 minus 𝑥 plus 12 equals zero.

Okay, so now, the final stage is to simplify the quadratic, and we do this by collecting like terms. So, when we do this, what we’re gonna get is 𝑥 squared minus 13𝑥 plus 12 equals zero. So now, if we go back to our original equation, what we can see is that the coefficient of 𝑥 is our 𝑚 and the numerical value on its own is our 𝑛. So therefore, we can say the value of 𝑚 is equal to negative 13 and the value of 𝑛 is equals 12.

Okay, great. We did mention that we could have two different methods to solve this problem. So, this is the first method. We’re now gonna take a look at another method. Well, for the second method, what we do is we take a look at our quadratic, and we notice that it’s in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And what we have is a special set of relationships to deal with the roots of our equation. Well, the two relationships we’re looking at are 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, as we already pointed out in method one, our roots are one and 12. And then, what we can also identify is our 𝑎, 𝑏, and 𝑐. Well, 𝑎 is one because we’ve got single 𝑥 squared, 𝑏 is equal to 𝑚, and 𝑐 is equal to 𝑛. So, therefore, using our first relationship which is the sum of the roots is equal to negative 𝑏 over 𝑎, we can say that one plus 12 equals negative 𝑚 over one. So therefore, we can say that 13 is equal to negative 𝑚. So then if we divide through by negative one, what we’re gonna get is negative 13 equals 𝑚, which is what we got with our first method. So great, we found 𝑚. So now, let’s find 𝑛.

Well, to find 𝑛, we’re gonna use the second relationship and that is that the product of the roots is equal to 𝑐 over 𝑎. Well, this is gonna give us one multiplied by 12 equals 𝑛 over one. So therefore, we can say that 12 is equal to 𝑛. So again, this is the same as we got in our first method. So therefore, we can confirm that the values of 𝑚 and 𝑛 are negative 13 and 12, respectively.