### Video Transcript

Given that ๐ฟ and ๐ are the roots
of the equation three ๐ฅ squared plus 16๐ฅ minus one equals zero, find, in its
simplest form, the quadratic equation whose roots are ๐ฟ over two and ๐ over
two.

Letโs 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 ๐ฅ 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 ๐ฅ squared, is three. And so, weโre going to divide every
single term by three. Three ๐ฅ squared divided by three
is ๐ฅ squared. 16๐ฅ divided by three is 16๐ฅ over
three or 16 over three ๐ฅ. 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 ๐ฅ. So, itโs negative 16 over
three. Then, the product is negative
one-third. But weโre also told that ๐ฟ and ๐
are the roots of our equation, so we can replace these with ๐ฟ plus ๐ as the sum
and ๐ฟ times ๐ as the product. Weโre looking to find the quadratic
equation whose roots are ๐ฟ over two and ๐ over two. And so, what weโre going to do is
find an expression for the sum of these roots; thatโs ๐ฟ over two plus ๐ over
two. Similarly, weโre going to find the
product of these roots; thatโs ๐ฟ over two times ๐ over two which is ๐ฟ๐ over
four.

So, how can we link the equations
we have? Well, letโs call this first
equation one. We have ๐ฟ plus ๐ as being equal
to negative 16 over three. If we divide the entire expression
by two, that is ๐ฟ plus ๐ over two, we actually know thatโs equal to ๐ฟ over two
plus ๐ over two. And so, this means we can find the
value of ๐ฟ over two plus ๐ over two by dividing the value for ๐ฟ plus ๐ by
two. Thatโs 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 ๐ฟ๐
divided by four. So thatโs 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 ๐ฟ over two and ๐ over two is ๐ฅ squared minus
negative eight over three ๐ฅ plus negative one twelfth equals zero.

Letโs 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, weโre going to
multiply through by 12. ๐ฅ squared times 12 is 12๐ฅ
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
12๐ฅ squared plus 32๐ฅ minus one equals zero.