Question Video: Evaluating Algebraic Expressions Using the Factorization of a Difference of Two Squares Mathematics

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.