Find two integers whose sum is -1 and product is -20

Answer 1

x + y = -1

xy = -20

y = -x - 1

x(-x - 1) = -20

-x² - x = - 20

x² + x - 20 = 0

(x + 5)(x - 4) = 0

What are the numbers?

Answer 2

Let's start by deciding how many and what types of variables we will need to solve this. The problem states that we should find two integers, so we will need two variables. Let's use m and n, which are common variable names when we talk about integers. Now, we need to convert the words of the problem into equations we can solve.

"Two integers whose sum is -1":

m + n = -1 (since sum is just another word for add)

"...and product is -20":

m * n = mn = -20 (we often omit the multiplication operator when using variables)

So, now we have two equations and two variables between them. This is an important condition to satisfy when solving systems of equations (our's is a system of two equations). In general, you must have the number of equations equal to the number of variables (unknowns) to solve for all variables.

Let me write the equations again next to each other for clarity:

m + n = -1

mn = -20

If you ever go on to take linear algebra you will use matrices to solve these types of systems, but here simple substitution will work just fine. The key is to isolate one of the variables, it does not matter which, but one is usually easier than the other. Here, I would isolate n in the first equation to get:

n = -1 - m

Now you can substitute n in the second equation with the n above:

m(-1 - m) = -20

By distributing the m we get:

-m - m² = -20

-m² - m + 20 = 0 (I rearranged some terms and did some addition--if you don't follow just ask me)

m² + m - 20 = 0 (I divided both sides by -1)

Now it's a matter of factoring a degree two polynomial. Since I have been doing this for a very long time I can see that it factors as:

(m + 5)(m-4) = 0

but do not worry if you can't see that right away. You will soon enough.

So that leads to

m = -5 or m = 4

**BEWARE** We are not yet finished. We have two integers but they are only candidates for one variable--m. We need to substitute each m back into one of the equations to get the value of n. Choose the simplest:

mn = -20

(-5)n = -20

n=4

or

(4)n = -20

n = -5

In our problem, no matter which m you choose (-5 or 4) you get a value of n that is the one not chosen. This is not always the case!

So to answer the question (which is *always* what we want to do), -5 and 4 are the two integers whose sum is -1 and product is -20.

Let me know if you would like clarification about anything I have written.

Christian

Answer 3

You do this type of problem by first identifying the factors of -20:

1 x -20

-1 x 20

2 x -10

-2 x 10

5 x -4

-5 x 4

There, that's all of them. Now which of these, if added, would equal -1?