The product of two consecutive positive even integers is 48. Find… Step by Step

Step by step

The question

Question: The product of two consecutive positive even integers is 48.

Find the integers.

1 Find what we know

What do we know?

Two even numbers in a row, 2 apart.

Times each other, they make 48.

What do we want?

The two numbers.

2 Name the unknown

We call the first number x.

x = first number

The next one is 2 more: x + 2.

x + 2 = second number

3 Write the equation

In words: first times second is the product.

x(x + 2) = 48

So the equation is x(x + 2) = 48.

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

Now we solve it.

x(x + 2) = 48

We open the bracket: x² + 2x = 48.

x² + 2x = 48

4 Get 0 on one side

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

So we take away 48 on both sides.

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

x² + 2x − 48 = 48 − 48

Now we have x² + 2x − 48 = 0.

x² + 2x − 48 = 0

5 Factor

We need two numbers.

They multiply to −48, and they add to 2.

? × ? = −48

? + ? = 2

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 48, smallest first.

We take the smaller from the bigger.

We need 2.

6 and 8 give 2.

Yes!

With the signs: −6 and 8.

1 × 48: 48 − 1 = 47

2 × 24: 24 − 2 = 22

3 × 16: 16 − 3 = 13

4 × 12: 12 − 4 = 8

6 × 8: 8 − 6 = 2 ✓

So it is (x − 6)(x + 8) = 0.

(x − 6)(x + 8) = 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 − 6 = 0 or x + 8 = 0

In x − 6 = 0, we add 6 on both sides.

In x + 8 = 0, we take away 8 on both sides.

x − 6 + 6 = 0 + 6 or x + 8 − 8 = 0 − 8

So x = 6 or x = −8.

x = 6 or x = −8

7 Answer the question

x = −8: the story says positive.

So we throw it away.

x = −8 ✗

the story says positive

Careful!

−8 and −6 also multiply to 48.

But the story asks for positive numbers.

Careful: the story says positive

x = 6 works: positive.

x = 6 ✓

positive

The second number is 6 plus 2: 8.

x + 2 = 6 + 2 = 8

The numbers are 6 and 8.

8 Check

Let's check in the story.

6 times 8 is 48, the product.

6 × 8 = 48

And they are even numbers in a row.

So the answer is right!

6, 8: even, 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