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
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 negative 48.
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