Video Transcript
Given that 𝐿 and 𝑚 are the roots
of the equation three 𝑥 squared plus 16𝑥 minus one equals zero, find, in its
simplest form, the quadratic equation whose roots are 𝐿 over two and 𝑚 over
two.
Let’s begin by recalling the
relationship between the quadratic equation and its roots. For a quadratic equation whose
leading coefficient is one, we can say that the coefficient of 𝑥 is the negative
sum of the roots of the equation, whereas the constant is the product of its
roots. Now, comparing this to our
equation, we see we do have a bit of a problem. The leading coefficient, the
coefficient of 𝑥 squared, is three. And so, we’re going to divide every
single term by three. Three 𝑥 squared divided by three
is 𝑥 squared. 16𝑥 divided by three is 16𝑥 over
three or 16 over three 𝑥. And then, our constant term becomes
negative one-third.
And so, comparing this to the
general form, we know that the sum of the roots of our equation is the negative
coefficient of 𝑥. So, it’s negative 16 over
three. Then, the product is negative
one-third. But we’re also told that 𝐿 and 𝑚
are the roots of our equation, so we can replace these with 𝐿 plus 𝑚 as the sum
and 𝐿 times 𝑚 as the product. We’re looking to find the quadratic
equation whose roots are 𝐿 over two and 𝑚 over two. And so, what we’re going to do is
find an expression for the sum of these roots; that’s 𝐿 over two plus 𝑚 over
two. Similarly, we’re going to find the
product of these roots; that’s 𝐿 over two times 𝑚 over two which is 𝐿𝑚 over
four.
So, how can we link the equations
we have? Well, let’s call this first
equation one. We have 𝐿 plus 𝑚 as being equal
to negative 16 over three. If we divide the entire expression
by two, that is 𝐿 plus 𝑚 over two, we actually know that’s equal to 𝐿 over two
plus 𝑚 over two. And so, this means we can find the
value of 𝐿 over two plus 𝑚 over two by dividing the value for 𝐿 plus 𝑚 by
two. That’s negative 16 over three
divided by two, which is negative eight over three.
And we can repeat this process with
our second equation. This time, of course, we want 𝐿𝑚
divided by four. So that’s going to be negative
one-third divided by four, which is negative one twelfth. And now that we have the sum of our
roots and the product, we can substitute these back into our earlier equation. When we do, we find that the
quadratic equation whose roots are 𝐿 over two and 𝑚 over two is 𝑥 squared minus
negative eight over three 𝑥 plus negative one twelfth equals zero.
Let’s simplify a little by dealing
with our signs. In other words, negative negative
eight-thirds is just eight-thirds. And adding negative one twelfth is
the same as subtracting one twelfth. Our final step is to create integer
coefficients. And to do so, we’re going to
multiply through by 12. 𝑥 squared times 12 is 12𝑥
squared. Then, if we multiply eight-thirds
by 12, we cancel a three. And we end up working out eight
times four, which is 32. Negative one twelfth times 12 is
negative one. And, of course, zero times 12 is
zero. And so, the quadratic equation is
12𝑥 squared plus 32𝑥 minus one equals zero.
