I’m thinking of two numbers. Use the clues to determine what the numbers are.
The product of the numbers is 48. And the difference between the numbers is two.
Let’s let one of the numbers be 𝑥, and one of the numbers be 𝑦. Our keywords are: product and
difference. Product means to multiply, and difference means to subtract. So if the product of
the numbers is 48, then 𝑥 times 𝑦 is equal to 48. And if the difference between the two
numbers is two, then 𝑥 minus 𝑦 equals two. Or, you could’ve put 𝑦 minus 𝑥 equals two.
Now we’re going to assume that our numbers are positive because this was actually a multiple
choice question and all of the numbers were positive, because we could actually have negative
integers as well. But again, we’re going to assume that they’re positive. So the first thing
that we need to do is to somehow take these two equations and use them together. So the best
idea would be to take one of the equations and solve for a variable.
So let’s take the second equation and solve for 𝑥. And we will do that by adding 𝑦 to both
sides of the equation. So now we know that 𝑥 is equal to 𝑦 plus two. So now we can take 𝑦
plus two and plug that in for 𝑥 in our first equation. So instead of 𝑥 times 𝑦 equals 48,
we’ll have 𝑦 plus two times 𝑦 equals 48. And we will use the distributive property, so we
have 𝑦 squared plus two 𝑦 equals 48.
Now in order to solve, we’re gonna have to factor. So we need to bring the 48 over to the
left-hand side of the equation by subtracting. So now we’ll have to find two numbers that
multiply to be negative 48 and add to be positive two.
So let’s first begin by listing numbers that multiply to be 48. Now, out of all of these
sets, which could we add and make positive two? Well, before we think about that, we have to
remember that these two numbers multiply to be negative 48. So one of the numbers will have
to be negative. We don’t know which one, but we know that they need to add to be positive two.
So if we would take eight and six, one of them would need to be negative and make a positive
two when they were added together. So since six is smaller, six would have to be negative.
So eight plus negative six is equal to positive two, and eight times negative six is equal to
So now our factors are 𝑦 plus eight and 𝑦 minus six. And now we need to say each factor equal
to zero. And now we need to solve both factors for 𝑦. So we need to subtract eight on
the one equation and add six on the other equation. So we get 𝑦 equals negative eight, and
we also get 𝑦 equals six. So like we stated before, we’re using the positive number. So 𝑦 is
equal to six. So if 𝑦 is equal to six, we can take that and plug it in and solve for 𝑥. So
instead of 𝑥 equals 𝑦 plus two, we’ll have 𝑥 equals six plus two. So 𝑥 is equal to eight.
So our two numbers are eight and six.