Video Transcript
If 𝐿 and 𝑚 are the roots of the
equation 𝑥 squared plus 20𝑥 plus 15 equals zero, what is the value of one over 𝑚
plus one over 𝐿?
We begin by reminding ourselves
what the relationship between the coefficient of a quadratic equation is and its
roots. For a quadratic equation whose
leading coefficient is one, in other words, the coefficient of 𝑥 squared is one,
the negative coefficient of 𝑥 tells us the sum of the roots and the constant term
tells us the product. And this is really useful because
the coefficient of 𝑥 here is 20 and the constant is 15. And so, the sum of our roots must
be negative 20. Remember, we said that it’s the
negative coefficient of 𝑥. Then, the product, which is the
constant term, must be 15.
But of course, we were told that 𝐿
and 𝑚 are the roots of our equation. So we can, in fact, say that 𝐿
plus 𝑚 must be negative 20 and 𝐿 times 𝑚 must be 15. So, how does this help? We’re looking to find the value of
one over 𝑚 plus one over 𝐿, and we can’t easily find two numbers that have a sum
of negative 20 and a product of 15. So we’re going to need to
manipulate our expressions. Let’s think of one over 𝑚 plus one
over 𝐿.
We know that to add two fractions,
we need to create a common denominator. Now, the easiest way to do this
when working with algebraic fractions is to multiply both parts of each fraction by
the denominator of the other. So, we’re going to multiply the
numerator and denominator of our first fraction by 𝐿 and of our second fraction by
𝑚. That gives us 𝐿 over 𝐿𝑚 plus 𝑚
over 𝐿𝑚. And now since the denominators are
the same, we simply add the numerators, and that gives us 𝐿 plus 𝑚 over 𝐿𝑚.
And this is really useful because
we know that the numerator 𝐿 plus 𝑚 is equal to negative 20, and then the
denominator 𝐿𝑚 is 15. And this means, in turn, that one
over 𝑚 plus one over 𝐿 is equal to negative 20 over 15, which simplifies to
negative four-thirds. And so, if 𝐿 and 𝑚 are the roots
of our equation, then one over 𝑚 plus one over 𝐿 must be equal to negative
four-thirds.
