Video Transcript
If 𝑥 times 𝑦 equals eight, what
is the value of 𝑥 plus three 𝑦 squared minus 𝑥 minus three 𝑦 squared?
In this question, we need to
determine the value of an algebraic expression by using the given fact that 𝑥 times
𝑦 equals eight. We cannot directly evaluate this
expression since we do not know the exact values of 𝑥 and 𝑦. Instead, we can look at the
expression we are asked to evaluate and note that it is the difference of two
squares. This means that we can factor this
expression by recalling that 𝑎 squared minus 𝑏 squared equals 𝑎 minus 𝑏 times 𝑎
plus 𝑏. Substituting 𝑎 equals 𝑥 plus
three 𝑦 and 𝑏 equals 𝑥 minus three 𝑦 into the difference of squares formula
allows us to factor the expression to obtain 𝑥 plus three 𝑦 minus 𝑥 minus three
𝑦 times 𝑥 plus three 𝑦 plus 𝑥 minus three 𝑦.
We can simplify this expression by
first distributing the negative over the parentheses to get the following
expression. We can now combine the like terms
in each factor to obtain six 𝑦 times two 𝑥. We can then rearrange the product
and use the fact that six times two is 12 to get 12𝑥𝑦. We are told in the question that 𝑥
times 𝑦 is eight, so we can substitute this in to obtain 12 times eight, which we
can calculate is 96.
This is enough to answer this
question. However, it is worth noting that we
have shown that the value of this expression does not change regardless of which
values of 𝑥 and 𝑦 we choose so long as their product is eight. This means that we can check our
answer by substituting any of these such values into the original expression. For instance, we can substitute 𝑥
equals two and 𝑦 equals four into the expression to obtain two plus three times
four squared minus two minus three times four squared. If we evaluate this expression, we
get 96, which agrees with our general calculation. We will always get 96 if the
product of 𝑥 and 𝑦 is eight.
