The product of two consecutive positive even integers is 48. Find the two positive #'s.

Question

I never really understood what integers were 

Answer 1

Hi Mary,

 

Try thinking of Integers like this: 

 

Pretend you are standing on the number line at Zero. There are lots of numbers (including fractions and decimals like .10, .25, .68, .99) between Zero and One, but you jump down the number line landing only landing on the stepping stones that are located exactly one unit away from each other.... so from Zero to One, One to Two, Two to Three, and so on. You may also go in the other direction from Zero to Negative One, from Negative One to Negative Two, and so on. 

 

Formally, the Set of Integers (Z) is a subset of the set of Real Numbers represented by {...,-3, -2, -2, -1, 0, 1, 2, 3,...}.

 

So, this question is really asking you to think of two positive even numbers you can multiply together to get 48, but they have to be consecutive even numbers. I'll bet you could come up with them.... 6 and 8! But,.... here is how you do this algebraically.

 

Think of a list of consecutive integers like... 3, 4, 5, 6, 7, 8.  Let's talk about that 4. It is a positive, even integer. What would I need to do to that 4 to get to the next integer, which is 5? Just add one, right? But what would I need to do to that 4 to get to the next positive EVEN integer, which is 6? I would need to add TWO!

 

So, let's pretend we don't know that the answer to this question is 6 and 8. 

 

Let First Integer = x

 

Let the next consecutive, EVEN integer = x + 2

 

Now, the question states that the product (which implies we have to multiply) is 48.

 

So, (x)(x + 2) = 48

 

Distribute the x.

 

x² + 2x = 48

 

Subtract 48 from both sides of the equation.

 

x² + 2x - 48 = 0

 

Factor.

 

(x + 8)(x - 6) = 0

 

Set both factors equal to zero.

 

x + 8 = 0

 

Subtract 8 from both sides of the equation.

 

x = -8 

 

But wait!!! The question says we are looking for POSITIVE even integers. - 8 is not POSITIVE!! We can't use that! Forget about it!!!

 

but, set the other factor equal to zero.

 

x - 6 = 0

 

Add 6 to both sides of the equation.

 

x = 6

 

Ding! Ding! Ding! We have a winner!!! 

 

hmmmm.... but only one...? 

 

Go back to the beginning. 

 

Let First Integer = x. That's our 6.

 

Let next Consecutive, EVEN integer = x + 2

 

Replace the x here with 6.

 

The next consecutive EVEN integer = x + 2 = 6 + 2 = 8

 

AHA!

 

Now we have both the 6 and the 8!

 

Good luck!