Completely factor 𝑥 squared minus 15𝑥𝑦 plus 54𝑦 squared.
It’s important to notice that we have 𝑥s and we have 𝑦s. So our factorised answer will have 𝑥s and 𝑦s. So inside the factored form, the 𝑥 will be first. And then the 𝑦 will be second just like in the expression.
So in order to find the numbers that go with these, we need to find two numbers that multiply to be 54 and add to be negative 15. Negative nine and negative six multiply to be positive 54. And negative nine plus negative six adds to be negative 15. And that’s exactly what we needed. So we need a negative nine and a negative six. So 𝑥 minus nine 𝑦 times 𝑥 minus six 𝑦 will be our final answer.
A way to check is to redistribute, remultiply and make sure that we get the expression we started with. 𝑥 times 𝑥 is 𝑥 squared. 𝑥 times negative six 𝑦 is negative six 𝑥𝑦. Keeping our variables in alphabetical order, negative nine 𝑦 times 𝑥 is negative nine 𝑥𝑦. And negative nine 𝑦 times negative six 𝑦 is equal to 54𝑦 squared. And simplifying this, we get 𝑥 squared minus 15𝑥𝑦 plus 54𝑦 squared. And that is indeed the expression we began with.
Therefore, as we said before, 𝑥 minus nine 𝑦 times 𝑥 minus six 𝑦 will be our final answer.