Step by step
The question
Question: choose a method and solve 2x² + 7x + 3 = 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 = 7: 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 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 plus number, so both are plus.
We list the pairs that make 6, smallest first.
We add each pair.
We need 7.
1 and 6 give 7.
Yes!
3 Split the middle
We write 7x 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 3.
Both leave the same bracket.
Now both parts have 2x + 1 in them.
5 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).
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 2x + 1 = 0, we take away 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.
7 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: 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.