Solve 2 cos^2 x + 3 cos x + 1 = 0 Step by Step

Step by step

The question

Question: solve 2 cos² x + 3 cos x + 1 = 0.

Find every angle from 0° up to 360°.

2 cos² x + 3 cos x + 1 = 0

We want the angles from 0° to 360°: one full turn.

0° ≤ x < 360°

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.

u = cos x

Then the equation is 2u² + 3u + 1 = 0: a quadratic.

We solve it the usual way.

2u² + 3u + 1 = 0

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 1: that is 2.

We need two numbers that multiply to 2 and add to 3.

2 × 1 = 2

? × ? = 2, ? + ? = 3

They multiply to a plus number, so the two numbers have the same sign.

They add to a plus number, so both are plus.

× is + → same signs: both +

We list the pairs that make 2, smallest first.

We add each pair.

We need 3.

1 and 2 give 3.

Yes!

1 × 2: 1 + 2 = 3 ✓

3 Split the middle

We write 3u as u + 2u.

It is the same amount.

2u² + u + 2u + 1 = 0

4 Group in pairs

We put the first two together, and the last two together.

(2u² + u) + (2u + 1) = 0

From the first pair we take out u.

From the second pair we take out 1.

Both leave the same bracket.

(2u² + u) = u(2u + 1)

(2u + 1) = 1(2u + 1)

Now both parts have 2u + 1 in them.

u(2u + 1) + 1(2u + 1) = 0

5 Same bracket out

2u + 1 is in both parts, so we take it out once.

What is left, u + 1, goes in the other bracket.

ab + cb = b(a + c)

So we get (2u + 1)(u + 1).

(2u + 1)(u + 1) = 0

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.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

2u + 1 = 0 or u + 1 = 0

In 2u + 1 = 0, we take away 1 on both sides.

In u + 1 = 0, we take away 1 on both sides.

2u + 1 − 1 = 0 − 1 or u + 1 − 1 = 0 − 1

Now we have 2u = −1 or u = −1.

2u = −1 or u = −1

To remove times 2, we divide both sides by 2.

So u = −1/2 or u = −1.

u = −12 or u = −1

7 Put cos x back

But u is cos x.

So cos x = −1 or cos x = −1/2.

cos x = −1 or −12

Careful!

These are not the answers yet!

They are values of cos x.

We still need the angles.

Careful: not x = −1 or −12 ✗

8 Find every angle

From the unit circle: cos 60° = 1/2.

So the reference angle is 60°.

The minus only changes which quadrants.

cos 60° = 12

Cos is the across value.

The points with that across value are on an up-and-down line.

The across value is −1 at one point only.

x = 180°.

cos x = −1: x = 180°

Cos is negative left of the y-axis: quadrants II and III.

180° − 60° and 180° + 60°: x = 120° or 240°.

cos x = −12: x = 120° or 240°

On the circle: 3 points.

All together, in order: x = 120°, 180°, 240°.

x = 120°, 180°, 240°

9 Check

Let's check every answer in 2 cos² x + 3 cos x + 1 = 0.

cos x is −1/2 at 120°.

Each one must give 0.

x = 120°: 2(−12)² + 3(−12) + 1 = 12 − 32 + 1 = 0

x = 180°: 2(−1)² + 3(−1) + 1 = 2 − 3 + 1 = 0

x = 240°: 2(−12)² + 3(−12) + 1 = 12 − 32 + 1 = 0

Every answer works.

So x = 120°, 180°, 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.

Answer