Video Transcript
Given that 𝐿 and 𝑚 are the roots
of the equation 𝑥 squared minus two 𝑥 plus five equals zero, find, in its simplest
form, the quadratic equation whose roots are 𝐿 squared and 𝑚 squared.
Let’s begin by recalling the
relationship between a quadratic equation whose leading coefficient is one and its
roots. We can represent it as 𝑥 squared
minus the sum of the roots times 𝑥 plus the product of the roots equals zero. And so, this is essentially saying
that if we have a quadratic equation equal to zero and the leading coefficient is
one, in other words, the coefficient of 𝑥 squared is one, the negative coefficient
of 𝑥 will tell us the sum of the roots and the constant term will tell us its
product.
So we take the equation 𝑥 squared
minus two 𝑥 plus five equals zero. The coefficient of 𝑥 is negative
two. And so the sum must be the negative
of negative two, so the sum of the roots must be positive two. Then the constant term is five. So the product of the roots must be
five. So can we find two numbers that
have a sum of two and a product of five? Well, no, not easily. We’re not going to get nice integer
solutions. And so instead, we’re going to form
equations using 𝐿 and 𝑚. Since the sum of our roots is two
and 𝐿 and 𝑚 are the roots, we can say 𝐿 plus 𝑚 must be equal to two. And then we can say that 𝐿 times
𝑚 is equal to five.
The roots of our new equation are
𝐿 squared and 𝑚 squared. And so, since their sum will be 𝐿
squared plus 𝑚 squared, we need to manipulate our equations to find an expression
for 𝐿 squared plus 𝑚 squared and another expression for their product, 𝐿 squared
𝑚 squared. Let’s label our equations as one
and two. We’re going to take the entirety of
equation one and, we’re going to square it. In other words, we square both
sides. So on the right-hand side, we get
two squared, which is, of course, equal to four. Then, on the left-hand side, we get
𝐿 plus 𝑚 squared, which we can consider to be 𝐿 plus 𝑚 times 𝐿 plus 𝑚.
And if we distribute these
parentheses, we get 𝐿 squared plus two 𝐿𝑚 plus 𝑚 squared equals four. And then, if we subtract two 𝐿𝑚
from both sides, we get the expression for the sum of the roots 𝐿 squared and 𝑚
squared. It’s four minus two 𝐿𝑚. But of course, we have an
expression for 𝐿𝑚; equation two tells us that 𝐿𝑚 is equal to five. And so, 𝐿 squared plus 𝑚 squared
becomes four minus two times five, which is four minus 10 or simply negative
six. So, we found the sum of the roots
of our new equation and, therefore, the negative coefficient of 𝑥.
We’re now going to repeat this
process for equation two; we’re going to square both sides. That is, 𝐿𝑚 squared equals five
squared. But, of course, five squared is
25. And we can distribute to the power
of two across both terms. And we get 𝐿 squared 𝑚 squared
equals 25. We, therefore, find that the sum of
our new roots is negative six and the product is 25. Let’s substitute these back into
the general form. When we do, we get 𝑥 squared minus
negative six 𝑥 plus 25 equals zero. And so, the quadratic equation
whose roots are 𝐿 squared and 𝑚 squared is 𝑥 squared plus six 𝑥 plus 25 equals
zero.
