The product of two consecutive positive integers is 30. Find the… Step by Step

Step by step

The question

Question: The product of two consecutive positive integers is 30.

Find the integers.

1 Find what we know

What do we know?

Two numbers in a row.

Times each other, they make 30.

What do we want?

The two numbers.

2 Name the unknown

We call the first number x.

x = first number

The next one is 1 more: x + 1.

x + 1 = second number

3 Write the equation

In words: first times second is the product.

x(x + 1) = 30

So the equation is x(x + 1) = 30.

There is x times x in it: a quadratic equation.

Now we solve it.

x(x + 1) = 30

We open the bracket: x² + x = 30.

x² + x = 30

4 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 30 on both sides.

We do it on both sides, so they stay equal.

x² + x − 30 = 30 − 30

Now we have x² + x − 30 = 0.

x² + x − 30 = 0

5 Factor

We need two numbers.

They multiply to −30, and they add to 1.

? × ? = −30

? + ? = 1

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 30, smallest first.

We take the smaller from the bigger.

We need 1.

5 and 6 give 1.

Yes!

With the signs: −5 and 6.

1 × 30: 30 − 1 = 29

2 × 15: 15 − 2 = 13

3 × 10: 10 − 3 = 7

5 × 6: 6 − 5 = 1 ✓

So it is (x − 5)(x + 6) = 0.

(x − 5)(x + 6) = 0

6 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

x − 5 = 0 or x + 6 = 0

In x − 5 = 0, we add 5 on both sides.

In x + 6 = 0, we take away 6 on both sides.

x − 5 + 5 = 0 + 5 or x + 6 − 6 = 0 − 6

So x = 5 or x = −6.

x = 5 or x = −6

7 Answer the question

x = −6: the story says positive.

So we throw it away.

x = −6 ✗

the story says positive

Careful!

−6 and −5 also multiply to 30.

But the story asks for positive numbers.

Careful: the story says positive

x = 5 works: positive.

x = 5 ✓

positive

The second number is 5 plus 1: 6.

x + 1 = 5 + 1 = 6

The numbers are 5 and 6.

8 Check

Let's check in the story.

5 times 6 is 30, the product.

5 × 6 = 30

And they are in a row.

So the answer is right!

5, 6: in a row

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

Answer