The length of a rectangle is 3 meters more than its width. The area… Step by Step

Step by step

The question

Question: The length of a rectangle is 3 meters more than its width.

The area is 40 square meters.

Find the length and the width.

1 Find what we know

What do we know?

The length is 3 more than the width.

The area is 40 square meters.

Area = length times width.

What do we want?

The length and the width.

2 Name the unknown

We know the length from the width.

So we call the width x.

x = width

The length is x + 3.

x + 3 = length

Let's draw it.

The long side is x + 3.

The short side is x.

3 Write the equation

In words: width times length is the area.

x(x + 3) = 40

So the equation is x(x + 3) = 40.

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

Now we solve it.

x(x + 3) = 40

We open the bracket: x² + 3x = 40.

x² + 3x = 40

4 Get 0 on one side

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

So we take away 40 on both sides.

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

x² + 3x − 40 = 40 − 40

Now we have x² + 3x − 40 = 0.

x² + 3x − 40 = 0

5 Factor

We need two numbers.

They multiply to −40, and they add to 3.

? × ? = −40

? + ? = 3

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 40, smallest first.

We take the smaller from the bigger.

We need 3.

5 and 8 give 3.

Yes!

With the signs: −5 and 8.

1 × 40: 40 − 1 = 39

2 × 20: 20 − 2 = 18

4 × 10: 10 − 4 = 6

5 × 8: 8 − 5 = 3 ✓

So it is (x − 5)(x + 8) = 0.

(x − 5)(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 − 5 = 0 or x + 8 = 0

In x − 5 = 0, we add 5 on both sides.

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

x − 5 + 5 = 0 + 5 or x + 8 − 8 = 0 − 8

So x = 5 or x = −8.

x = 5 or x = −8

7 Answer the question

x = −8: a length cannot be negative.

So we throw it away.

x = −8 ✗

a length cannot be negative

Careful!

Both numbers make the equation true.

But only one makes sense in the story.

Always look back at the story.

Careful: x = −8 fits the equation, not the story

x = 5 works: a width.

x = 5 ✓

a width

The length is x + 3: 5 plus 3 is 8.

length = 5 + 3 = 8

The width is 5 m and the length is 8 m.

8 Check

Let's check in the story.

Length times width: 8 times 5 is 40, the area.

8 × 5 = 40

And 8 is 5 plus 3, like the story says.

So the answer is right!

8 = 5 + 3

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