Step by step
The question
Question: The product of two consecutive positive odd integers is 63.
Find the integers.
1 Find what we know
What do we know?
Two odd numbers in a row, 2 apart.
Times each other, they make 63.
What do we want?
The two numbers.
2 Name the unknown
We call the first number x.
The next one is 2 more: x + 2.
3 Write the equation
In words: first times second is the product.
So the equation is x(x + 2) = 63.
There is x times x in it: a quadratic equation.
Now we solve it.
We open the bracket: x² + 2x = 63.
4 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 63 on both sides.
We do it on both sides, so they stay equal.
Now we have x² + 2x − 63 = 0.
5 Factor
We need two numbers.
They multiply to −63, and they add to 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.
We list the pairs that make 63, smallest first.
We take the smaller from the bigger.
We need 2.
7 and 9 give 2.
Yes!
With the signs: −7 and 9.
So it is (x − 7)(x + 9) = 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 + 9 = 0, we take away 9 on both sides.
So x = 7 or x = −9.
7 Answer the question
x = −9: the story says positive.
So we throw it away.
Careful!
−9 and −7 also multiply to 63.
But the story asks for positive numbers.
x = 7 works: positive.
The second number is 7 plus 2: 9.
The numbers are 7 and 9.
8 Check
Let's check in the story.
7 times 9 is 63, the product.
And they are odd numbers 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.