The product of two consecutive integers is 5 more than their sum. Find the integers.

Question

The product of two consecutive integers is 5 more than their sum. Find the integers.

Answer 1

These types of questions ask you to make an equation then solve it.

For this one we need to find two numbers.

Consecutive integers means two numbers that are next to each other like 2 and 3.

These numbers can be written as n for the first one and n + 1 for the second.

From the question we need to find their product and sum.

So their product is

n * (n+1)

Using distribution their product is

n² + n product

Their sum will be

n + (n+1)

= n+n+1

=2n + 1 sum

Now we create an equation.

The product is 5 more than the sum.

5 more means that one side needs to have a + 5

Choosing the right side is difficult, but remember that the word is usually is the same as the equals sign.

Since 5 more is after the word is,

the product is the same as their sum + 5.

Writing the full equation we get.

n² +n + = 2n + 1 + 5

This is not too difficult to solve.

First move all numbers to the left side.

n² +n + = 2n + 6

-2n - 6 -2n - 6

n² - n - 6= 0

n can be found by factoring or using the quadratic equation.

I will use factoring.

(n - 3) * (n + 2) = 0

n = 3 , -2 Both of these can be the value of n.

Remember we decided the integers were n and n + 1.

Now it is time to plug in the values of n. One at a time.

For n = 3

The integers are 3 and 4

For n = -2

The integers are -2 and -1

Since an integer is any whole number positive or negative,

both are the answer

so both should be included in your answer.

You can check these too.

3*4 =(3 + 4) + 5

(-2)*(-1) = (-2 + (-1)) + 5

Answer 2

Two consecutive integers: x, x + 1

Product = x(x+1) = x^2 + x

Sum = x + x + 1 = 2x + 1

Product = 5 more than their sum:

x^2 + x = 2x + 1 + 5

x^2 + x = 2x + 6

x^2 -x -6 = 0

(x-3)(x+2) = 0

x either equals 3 or -2

so 2 consecutive integers either 3, 4 or -2, - 1

Let's check. product of 3 and 4 = 12 which is 5 more than their sum 7.

product of -2 and -1 = 2 which is 5 more than their sum -3

So answer is 3,4 or -1,-2