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.
