Step by step
The question
Question: solve 2x² − 7x + 3 = 0.
1 Find two numbers
There is a number in front of x², but it can still factor.
The number in front of x² is 2.
So we multiply 2 by 3: that is 6.
We need two numbers that multiply to 6 and add to −7.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 6, smallest first.
We add each pair.
We need 7.
1 and 6 give 7.
Yes!
With the signs: −1 and −6.
2 Split the middle
We write −7x as −x − 6x.
It is the same amount.
3 Group in pairs
We put the first two together, and the last two together.
From the first pair we take out x.
From the second pair we take out −3.
Both leave the same bracket.
Now both parts have 2x − 1 in them.
4 Same bracket out
2x − 1 is in both parts, so we take it out once.
What is left, x − 3, goes in the other bracket.
So we get (2x − 1)(x − 3).
5 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 2x − 1 = 0, we add 1 on both sides.
In x − 3 = 0, we add 3 on both sides.
Now we have 2x = 1 or x = 3.
To remove times 2, we divide both sides by 2.
So x = 1/2 or x = 3.
6 Check
Let's check: put x = 1/2 into the left side.
We get 0, the same as the right side.
It works!
Now put x = 3 into the left side.
We get 0 again, the same as the right side.
So both answers are right!
Let's look back.
First: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Last: check.