Step by step
The question
Question: The length of a rectangle is 5 feet more than its width.
The area is 84 square feet.
Find the length and the width.
1 Find what we know
What do we know?
The length is 5 more than the width.
The area is 84 square feet.
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.
The length is x + 5.
Let's draw it.
The long side is x + 5.
The short side is x.
3 Write the equation
In words: width times length is the area.
So the equation is x(x + 5) = 84.
There is x times x in it: a quadratic equation.
Now we solve it.
We open the bracket: x² + 5x = 84.
4 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 84 on both sides.
We do it on both sides, so they stay equal.
Now we have x² + 5x − 84 = 0.
5 Factor
We need two numbers.
They multiply to −84, and they add to 5.
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.
We list the pairs that make 84, smallest first.
We take the smaller from the bigger.
We need 5.
7 and 12 give 5.
Yes!
With the signs: −7 and 12.
So it is (x − 7)(x + 12) = 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.
We make each bracket 0.
Two small equations.
In x − 7 = 0, we add 7 on both sides.
In x + 12 = 0, we take away 12 on both sides.
So x = 7 or x = −12.
7 Answer the question
x = −12: a length cannot be negative.
So we throw it away.
Careful!
Both numbers make the equation true.
But only one makes sense in the story.
Always look back at the story.
x = 7 works: a width.
The length is x + 5: 7 plus 5 is 12.
The width is 7 feet and the length is 12 feet.
8 Check
Let's check in the story.
Length times width: 12 times 7 is 84, the area.
And 12 is 7 plus 5, like the story says.
So the answer is right!
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.