Step by step
The question
Question: choose a method and solve 3x² − 5x − 2 = 0.
1 Choose the method
We ask in order.
First: is there no x term?
No, there is one.
Two whole numbers multiply to a × c = −6 and add to b = −5: 1 and −6.
So it factors.
So: factoring.
2 Find two numbers
There is a number in front of x², but it can still factor.
The number in front of x² is 3.
So we multiply 3 by −2: that is −6.
We need two numbers that multiply to −6 and add to −5.
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.
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 5.
1 and 6 give 5.
Yes!
With the signs: 1 and −6.
3 Split the middle
We write −5x as x − 6x.
It is the same amount.
4 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 −2.
Both leave the same bracket.
Now both parts have 3x + 1 in them.
5 Same bracket out
3x + 1 is in both parts, so we take it out once.
What is left, x − 2, goes in the other bracket.
So we get (3x + 1)(x − 2).
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 3x + 1 = 0, we take away 1 on both sides.
In x − 2 = 0, we add 2 on both sides.
Now we have 3x = −1 or x = 2.
To remove times 3, we divide both sides by 3.
So x = −1/3 or x = 2.
7 Check
Let's check: put x = −1/3 into the left side.
We get 0, the same as the right side.
It works!
Now put x = 2 into the left side.
We get 0 again, the same as the right side.
So both answers are right!
Let's look back.
First: choose the method.
Then: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Last: check.