Step by step
The question
Question: solve 2x² + 5x − 3 = 0.
There is an x².
1 Find two numbers
There is a number in front of x², but it can still factor.
The rule: we split the middle into two parts.
Their numbers must multiply to first × last.
Here the first number is 2 and the last is −3.
2 × −3 = −6.
So we need two numbers that multiply to −6.
They must also add to the middle number, 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.
When we add a plus and a minus, the bigger one wins.
We list the pairs that make 6, smallest first.
For each pair, we check two things: they multiply to 6, and the bigger take away the smaller is 5.
One is minus, so adding them is a take away.
1 × 6 = 6.
6 − 1 = 5.
Both work!
So the two numbers are 1 and 6.
With the signs: −1 and 6.
2 Split the middle
We write 5x as −x + 6x.
It is the same amount.
3 Group in pairs
We put the first two together, and the last two together.
2x² and −x both have x in them.
So from the first pair we take out x.
6x and −3 both have 3 in them.
So 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 take away 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.