### Video Transcript

If π₯ plus π¦ all squared is equal to 100
and π₯π¦ equals 20, what is the value of π₯ squared plus π¦ squared?

So, weβve been given two pieces of
information about these numbers π₯ and π¦ and asked to use them to determine the value of π₯
squared plus π¦ squared. Now, your first thought may be that π₯
plus π¦ all squared is just equal to π₯ squared plus π¦ squared. In which case, the value weβre looking
for is the value given in the question; itβs 100. But if this is the case, why have we also
been given the value of π₯π¦?

In fact, if we were to answer the
question this way, weβd have made one of the most common mistakes in mathematics because
weβve incorrectly expanded the binomial. Remember that π₯ plus π¦ all squared
means π₯ plus π¦ multiplied by π₯ plus π¦. So, we are, in fact, multiplying a
binomial by itself, not just squaring each of the individual terms.

Letβs see what happens if we correctly
expand π₯ plus π¦ all squared. Using the FOIL method which, remember,
stands for firsts, outers, inners, lasts, this gives π₯ squared plus π₯π¦ plus π₯π¦ plus π¦
squared, which simplifies to π₯ squared plus two π₯π¦ plus π¦ squared. What we now have is an equation
connecting π₯ plus π¦ all squared, whose value we know, π₯π¦, whose value we know, and π₯
squared plus π¦ squared, whose value we wish to calculate.

Substituting 100 for π₯ plus π¦ all
squared and 20 for π₯π¦, we have 100 equals π₯ squared plus π¦ squared plus two multiplied
by 20. That simplifies to 100 equals π₯ squared
plus π¦ squared plus 40. And subtracting 40 from each side, we
find that π₯ squared plus π¦ squared is equal to 60. So, weβve solved the problem. By correctly expanding the binomial π₯
plus π¦ all squared and then substituting the values given in the question, we found that π₯
squared plus π¦ squared is equal to 60.