The product of two consecutive integers is 13 less than five times their sum. Find the integers.

Question

I do not understand how to solve this word problem.

Answer 1

Don't show fear! Word problems can smell fear!

The key to all word problems is tidying it up and making sense of it. First, think about consecutive integers. The idea is simply that you find each new value by adding one to the previous value. In other words:

First           X

Second:      X + 1

Since "product" means "multiply", we do exactly that:

x(x+1) = 5 * (X + X + 1) - 13

Notice that we wrote "13" last? The magic words "than, from, into" always mean "write this part backward".

Let's simplify our messy equation:

X² + X = 5(2X + 1) - 13

X² + X = 10X + 5 -13

X² + X = 10X -8

Ah-ha! We have a quadratic, so we must always get a zero on one side:

X² + X -10X + 8 = 0

X² -9X + 8 =0

Now, I have to use the f-word: "factoring"!

(X - 8)(X - 1) = 0

Therefore, setting each factor to zero and solving, we get...

X = 8   ,  X = 1