Step by step
The question
Question: solve 3x² − 10x − 8 = 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 3 and the last is −8.
3 × −8 = −24.
So we need two numbers that multiply to −24.
They must also add to the middle number, −10.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
When we add a plus and a minus, the bigger one wins.
We list the pairs that make 24, smallest first.
For each pair, we check two things: they multiply to 24, and the bigger take away the smaller is 10.
One is minus, so adding them is a take away.
1 × 24 = 24.
24 − 1 = 23.
Not 10.
2 × 12 = 24.
12 − 2 = 10.
Both work!
So the two numbers are 2 and 12.
With the signs: 2 and −12.
2 Split the middle
We write −10x as 2x − 12x.
It is the same amount.
3 Group in pairs
We put the first two together, and the last two together.
3x² and 2x both have x in them.
So from the first pair we take out x.
−12x and −8 both have −4 in them.
So from the second pair we take out −4.
We take out the minus too, so the bracket matches the first one.
Both leave the same bracket.
Now both parts have 3x + 2 in them.
4 Same bracket out
3x + 2 is in both parts, so we take it out once.
What is left, x − 4, goes in the other bracket.
So we get (3x + 2)(x − 4).
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 3x + 2 = 0, we take away 2 on both sides.
In x − 4 = 0, we add 4 on both sides.
Now we have 3x = −2 or x = 4.
To remove times 3, we divide both sides by 3.
So x = −2/3 or x = 4.
6 Check
Let's check: put x = −2/3 into the left side.
We get 0, the same as the right side.
It works!
Now put x = 4 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.