Step by step
The question
Question: solve 2x² + 3x − 5 = 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 −5: that is −10.
We need two numbers that multiply to −10 and add to 3.
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.
We list the pairs that make 10, smallest first.
We take the smaller from the bigger.
We need 3.
2 and 5 give 3.
Yes!
With the signs: −2 and 5.
2 Split the middle
We write 3x as −2x + 5x.
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 2x.
From the second pair we take out 5.
Both leave the same bracket.
Now both parts have x − 1 in them.
4 Same bracket out
x − 1 is in both parts, so we take it out once.
What is left, 2x + 5, goes in the other bracket.
So we get (x − 1)(2x + 5).
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 x − 1 = 0, we add 1 on both sides.
In 2x + 5 = 0, we take away 5 on both sides.
Now we have x = 1 or 2x = −5.
To remove times 2, we divide both sides by 2.
So x = 1 or x = −5/2.
6 Check
Let's check: put x = 1 into the left side.
We get 0, the same as the right side.
It works!
Now put x = −5/2 into the left side.
We get 0 again, the same as the right side.
So both answers are right!
Let's look back.
First: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Last: check.