### Video Transcript

Given that πΏ and π are the roots
of the equation π₯ squared minus two π₯ plus five equals zero, find, in its simplest
form, the quadratic equation whose roots are πΏ squared and π squared.

Letβs begin by recalling the
relationship between a quadratic equation whose leading coefficient is one and its
roots. We can represent it as π₯ squared
minus the sum of the roots times π₯ 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 π₯ squared is one, the negative coefficient
of π₯ will tell us the sum of the roots and the constant term will tell us its
product.

So we take the equation π₯ squared
minus two π₯ plus five equals zero. The coefficient of π₯ 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. Weβre not going to get nice integer
solutions. And so instead, weβre going to form
equations using πΏ and π. Since the sum of our roots is two
and πΏ and π are the roots, we can say πΏ plus π must be equal to two. And then we can say that πΏ times
π is equal to five.

The roots of our new equation are
πΏ squared and π squared. And so, since their sum will be πΏ
squared plus π squared, we need to manipulate our equations to find an expression
for πΏ squared plus π squared and another expression for their product, πΏ squared
π squared. Letβs label our equations as one
and two. Weβre going to take the entirety of
equation one and, weβre 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
πΏ plus π squared, which we can consider to be πΏ plus π times πΏ plus π.

And if we distribute these
parentheses, we get πΏ squared plus two πΏπ plus π squared equals four. And then, if we subtract two πΏπ
from both sides, we get the expression for the sum of the roots πΏ squared and π
squared. Itβs four minus two πΏπ. But of course, we have an
expression for πΏπ; equation two tells us that πΏπ is equal to five. And so, πΏ squared plus π 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 π₯.

Weβre now going to repeat this
process for equation two; weβre going to square both sides. That is, πΏπ 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 πΏ squared π squared
equals 25. We, therefore, find that the sum of
our new roots is negative six and the product is 25. Letβs substitute these back into
the general form. When we do, we get π₯ squared minus
negative six π₯ plus 25 equals zero. And so, the quadratic equation
whose roots are πΏ squared and π squared is π₯ squared plus six π₯ plus 25 equals
zero.