Given that 𝑥 squared plus two 𝑥𝑦
plus 𝑦 squared is equal to 81, what are the possible values of 𝑥 plus 𝑦?
In this question, we are given an
equation involving two unknowns, 𝑥 and 𝑦. And we want to determine all of the
possible values of 𝑥 plus 𝑦 that satisfy the equation. To do this, we want to isolate 𝑥
plus 𝑦 on one side of the equation so that we can determine the possible values of
𝑥 plus 𝑦.
At first, it may seem difficult to
do this, since the left-hand side of the equation is a quadratic. However, we can note that the
quadratic is in the form of a perfect square by recalling that 𝑎 plus 𝑏 all
squared is equal to 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. If we set 𝑎 equal to 𝑥 and 𝑏
equal to 𝑦, then we see that 𝑥 plus 𝑦 all squared is equal to 𝑥 squared plus two
𝑥𝑦 plus 𝑦 squared. So we can factor the left-hand side
of the equation to obtain 𝑥 plus 𝑦 all squared. This is equal to 81.
We can solve for 𝑥 plus 𝑦 by
taking square roots of both sides of the equation. It is important to remember that
when we take the square root of both sides of an equation, we need to consider both
the positive and negative root. We have that 𝑥 plus 𝑦 equals
either positive or negative the square root of 81. Finally, we can calculate that the
square root of 81 is nine. So we have shown that 𝑥 plus 𝑦
must be equal to either nine or negative nine.