Step by step
The question
Question: The product of two consecutive positive integers is 56.
Find the integers.
1 Find what we know
What do we know?
Two numbers in a row.
Times each other, they make 56.
What do we want?
The two numbers.
2 Name the unknown
We call the first number x.
The next one is 1 more: x + 1.
3 Write the equation
In words: first times second is the product.
So the equation is x(x + 1) = 56.
There is x times x in it: a quadratic equation.
Now we solve it.
We open the bracket: x² + x = 56.
4 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 56 on both sides.
We do it on both sides, so they stay equal.
Now we have x² + x − 56 = 0.
5 Factor
We need two numbers.
They multiply to −56, and they add to 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.
We list the pairs that make 56, smallest first.
We take the smaller from the bigger.
We need 1.
7 and 8 give 1.
Yes!
With the signs: −7 and 8.
So it is (x − 7)(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.
We make each bracket 0.
Two small equations.
In x − 7 = 0, we add 7 on both sides.
In x + 8 = 0, we take away 8 on both sides.
So x = 7 or x = −8.
7 Answer the question
x = −8: the story says positive.
So we throw it away.
Careful!
−8 and −7 also multiply to 56.
But the story asks for positive numbers.
x = 7 works: positive.
The second number is 7 plus 1: 8.
The numbers are 7 and 8.
8 Check
Let's check in the story.
7 times 8 is 56, the product.
And they are in a row.
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.