If 𝑥 plus 𝑦 all squared is equal to 100
and 𝑥𝑦 equals 20, what is the value of 𝑥 squared plus 𝑦 squared?
So, we’ve been given two pieces of
information about these numbers 𝑥 and 𝑦 and asked to use them to determine the value of 𝑥
squared plus 𝑦 squared. Now, your first thought may be that 𝑥
plus 𝑦 all squared is just equal to 𝑥 squared plus 𝑦 squared. In which case, the value we’re looking
for is the value given in the question; it’s 100. But if this is the case, why have we also
been given the value of 𝑥𝑦?
In fact, if we were to answer the
question this way, we’d have made one of the most common mistakes in mathematics because
we’ve incorrectly expanded the binomial. Remember that 𝑥 plus 𝑦 all squared
means 𝑥 plus 𝑦 multiplied by 𝑥 plus 𝑦. So, we are, in fact, multiplying a
binomial by itself, not just squaring each of the individual terms.
Let’s see what happens if we correctly
expand 𝑥 plus 𝑦 all squared. Using the FOIL method which, remember,
stands for firsts, outers, inners, lasts, this gives 𝑥 squared plus 𝑥𝑦 plus 𝑥𝑦 plus 𝑦
squared, which simplifies to 𝑥 squared plus two 𝑥𝑦 plus 𝑦 squared. What we now have is an equation
connecting 𝑥 plus 𝑦 all squared, whose value we know, 𝑥𝑦, whose value we know, and 𝑥
squared plus 𝑦 squared, whose value we wish to calculate.
Substituting 100 for 𝑥 plus 𝑦 all
squared and 20 for 𝑥𝑦, we have 100 equals 𝑥 squared plus 𝑦 squared plus two multiplied
by 20. That simplifies to 100 equals 𝑥 squared
plus 𝑦 squared plus 40. And subtracting 40 from each side, we
find that 𝑥 squared plus 𝑦 squared is equal to 60. So, we’ve solved the problem. By correctly expanding the binomial 𝑥
plus 𝑦 all squared and then substituting the values given in the question, we found that 𝑥
squared plus 𝑦 squared is equal to 60.