Step by step
The question
Question: solve 2 cos² x + 5 cos x + 2 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Name it
Look: cos² x and cos x.
Like x² and x in a quadratic.
We call cos x by one letter: u = cos x.
Then the equation is 2u² + 5u + 2 = 0: a quadratic.
We solve it the usual way.
2 Find two numbers
There is a number in front of u², but it can still factor.
The number in front of u² is 2.
So we multiply 2 by 2: that is 4.
We need two numbers that multiply to 4 and add to 5.
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 4, smallest first.
We add each pair.
We need 5.
1 and 4 give 5.
Yes!
3 Split the middle
We write 5u as u + 4u.
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 u.
From the second pair we take out 2.
Both leave the same bracket.
Now both parts have 2u + 1 in them.
5 Same bracket out
2u + 1 is in both parts, so we take it out once.
What is left, u + 2, goes in the other bracket.
So we get (2u + 1)(u + 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 2u + 1 = 0, we take away 1 on both sides.
In u + 2 = 0, we take away 2 on both sides.
Now we have 2u = −1 or u = −2.
To remove times 2, we divide both sides by 2.
So u = −1/2 or u = −2.
7 Put cos x back
But u is cos x.
So cos x = −2 or cos x = −1/2.
Careful!
These are not the answers yet!
They are values of cos x.
We still need the angles.
8 Find every angle
cos x = −2 can never happen: across on the unit circle is always from −1 to 1.
No angle there.
From the unit circle: cos 60° = 1/2.
So the reference angle is 60°.
The minus only changes which quadrants.
Cos is the across value.
The points with that across value are on an up-and-down line.
Cos is negative left of the y-axis: quadrants II and III.
180° − 60° and 180° + 60°: x = 120° or 240°.
On the circle: 2 points.
All together, in order: x = 120°, 240°.
9 Check
Let's check every answer in 2 cos² x + 5 cos x + 2 = 0.
cos x is −1/2 at 120°.
Each one must give 0.
Every answer works.
So x = 120°, 240°.
Let's look back.
First: name it.
Then: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Then: put cos x back.
Then: find every angle.
Last: check.